What is true about Numerical Aperture?
t gives the minimum size that a microscope can resolve
it gives the maximum magnification for a telescope
it describes the opening of the cone of light that enters the objective
Light collected is proportional to NA
Values > 1 are impossible
values > 0.95 are rare for objectives working in air

a. The resistance of 100 meters of #18 AWG Copper wire at 20°C is 0.2098 Ω

b. To calculate the resistance of a wire, the cross-sectional area of the wire is required.

c. The area required to calculate the resistance is 2.155 × [tex]10^{-10}[/tex] m². The resistance of copper wire at 100°C is 92.2 Ω.

a. The resistance of 100 meters of #18 AWG Copper wire at 20°C can be determined using the formula;

R = ρL/A

A = πr²ρ

where;

R = resistance

ρ = resistivity

L = length of the wire

A = area of cross-section

r = radius of the wire

Substituting the given values;

Length of wire L = 100 meters

Area of cross-section A = ?

Diameter of wire d = 0.0403 inches or 1.02462 mm

Cross-sectional area A = πd²/4 = π(1.02462 mm)²/4 = 0.8231 mm²

Resistivity ρ = 1.724 x [tex]10^{-8}[/tex] Ω-m (at 20°C for copper)

Thus;

R = ρL/A = 1.724 x [tex]10^{-8}[/tex] Ω-m x 100 meters / 0.8231 mm²R = 0.2098 Ω

a. The resistance of 100 meters of #18 AWG Copper wire at 20°C is 0.2098 Ω

b. To calculate the resistance of a wire, the cross-sectional area of the wire is required.

c. To find the resistance of 600 meters of solid Copper wire with a diameter of 5 mm, we need to know the cross-sectional area of the wire. The formula for the cross-sectional area is;

A = πr²A = π(5/2)²A = 19.63 mm²

The resistivity of copper is 1.724 × [tex]10^{-8}[/tex] Ωm. Using the formula;

R = ρL/A

where;

L = 600 mA = 19.63 mm²

ρ = 1.724 × [tex]10^{-8}[/tex] Ωm

R = 0.16 ΩP.

To find the area required to calculate the resistance, the cross-sectional area of the wire is required. If the resistance of copper wire is 80 ohms at 20°C, we can use the above formula for resistivity.

ρ = RA/L

where;

R = 80 Ω

A = ?

L = 1 m

ρ = 1.724 × [tex]10^{-8}[/tex] Ωm

A = ρL/R = 1.724 × [tex]10^{-8}[/tex] × 1/80A = 2.155 × [tex]10^{-10}[/tex] m²

The resistance of copper wire at 100°C can be determined using the formula;

Rt = R0 [1 + α(T[tex]_{t}[/tex] - T[tex]_{0}[/tex])]

where;

R0 = resistance at 20°C = 80 Ω

T0 = temperature at 20°C = 293 K (20 + 273)

Tt = temperature at 100°C = 373 K (100 + 273)

α = temperature coefficient of copper = 0.00393/°C

Rt = 80 [1 + 0.00393(373 - 293)]R[tex]_{t}[/tex] = 92.2 Ω

Answer:

Therefore area required to calculate the resistance is 2.155 × [tex]10^{-10}[/tex] m². The resistance of copper wire at 100°C is 92.2 Ω.

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The range in wavelengths (in vacuum) for visible light in the frequency range between 7.9 × 10¹⁴ Hz (violet light) is 380 nm (approx).

The formula is given as:

frequency = (speed of light) / (wavelength)

Where:

frequency = 7.9 x 10¹⁴ Hz

speed of light = 3 x 10⁸ m/s (in vacuum)

Solving for wavelength:

wavelength = (speed of light) / (frequency)

Therefore, wavelength = (3 x 10⁸) / (7.9 x 10¹⁴) = 3.80 x 10⁻⁷ m or 380 nm (approx)

Hence, the range in wavelengths (in vacuum) for visible light in the frequency range between 7.9 × 10¹⁴ Hz (violet light) is 380 nm (approx).

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Starting from rest at the top of a frictionless inclined plane, a block takes 2 s to slide down to the bottom The incline angle is 0, where sin 0 = 3/4 and cos 0 = 2/3.  Thus, the length of the inclined plane is 20 m

The given incline angle is θ = 0 where sin θ = 3/4 and cos θ = 2/3 and the block slides down without any friction.

We are to find out the length of the inclined plane.

Let L be the length of the inclined plane, and g be the acceleration due to gravity.

As per the given statement, the block takes 2 seconds to slide down to the bottom of the inclined plane.

The acceleration of the block will be the same as the acceleration due to gravity in the direction of the inclined plane.

Therefore, the time t it takes for the block to slide down the incline plane of length L, starting from rest at the top of the inclined plane, is given by;         L = 1/2gt² (since initial velocity, u = 0)At θ = 0, sin θ = 3/4 and cos θ = 2/3.

Therefore, the length of the inclined plane is; L = 1/2 × 9.8 m/s² × (2 s)² = 19.6 m

Thus, the length of the inclined plane is 20 m (approximated to one significant figure).Hence, the correct option is (e) 20 m.

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The voltage across the resistor is 70 V.

Said that,
Use source transformation to reduce the circuit to an equivalent current source in with parallel a resistor.

Step 1: Convert the voltage source to a current source.

Isc = V/R

    = 60/30

    = 2 A

Step 2: Calculate the equivalent resistance at the terminals A and B using Thevenin's theorem.

R = 70 Ω//10 Ω + 40 Ω

  = 70 Ω//50 Ω

  = 35 Ω

Step 3: Find the current through the 35 Ω resistor using Ohm's law.

I = V/R

 = 2 A

Step 4: Find the voltage across the 35 Ω resistor using Ohm's law.

V = IR

  = 2 A × 35 Ω

  = 70 V

Therefore, the voltage across the resistor is 70 V.

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The ball rises to a height of 18.5 meters when it reaches a 25-degree incline.

When the 41 kg metal ball reaches a 25 degree incline, the height it goes to can be calculated. Here's how you can calculate the height of the ball:

First, we will calculate the potential energy of the ball by utilizing the formula: potential energy = mass * gravity * height

PE = mgh

Where m = 41 kg, g = 9.81 m/s² (the acceleration due to gravity), and h is the height in meters.

Since the ball is rolling at 19 m/s on a level surface, its kinetic energy will be:

kinetic energy = 0.5 * mass * velocity²

KE = 0.5 * m * v²

KE = 0.5 * 41 * 19²

KE = 7383.5 J

Now, we will equate the potential energy to the kinetic energy since the energy is conserved:

PE = KE => mgh = 7383.5Jh = 7383.5 / (41 * 9.81)h = 18.5 m

Therefore, the ball rises to a height of 18.5 meters when it reaches a 25-degree incline.

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The electric field at the fourth vacant corner is 4.05 × 10⁵ N/C.

Given,Three charged particles are located at the three corners of a rectangle.The magnitude of q1, q2 and q3 are given as 3 nC, 5 nC and 6 nC respectively.The value of x = 0.6m and the value of y = 0.2m.Figure 4.1The electric field at the fourth vacant corner can be calculated as follows:

We can make use of the formula given below to find the magnitude of the electric field,where k is the Coulomb constant and the magnitude of q1, q2 and q3 are given as 3 nC, 5 nC and 6 nC respectively, The value of x = 0.6m and the value of y = 0.2m. E = kq/r²Where k = 9 × 10⁹ N m²/C²The magnitude of q1, q2 and q3 are given as 3 nC, 5 nC and 6 nC respectively.r₁ = x² + y²r₁ = 0.6² + 0.2²r₁ = √(0.36 + 0.04)r₁ = √0.4r₁ = 0.6324 m r₂ = y²r₂ = 0.2²r₂ = 0.04 mTherefore, the electric field at the fourth vacant corner is 4.05 × 10⁵ N/C (approx).

Thus, the electric field at the fourth vacant corner is 4.05 × 10⁵ N/C.

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The power dissipated by the 20 ohm lamp is 0.5556 W and the power dissipated by the 30 ohm lamp is 0.8333 W.

Two lamps having resistances of 20 ohm and 30 ohm are connected in series with a 10V battery. The current in the circuit is given by:I = V/R (series circuit)Resistance of the circuit, R = R₁ + R₂I = 10/(20 + 30)I = 0.1667ANow, using Ohm's Law:Power dissipated by the 20 ohm lamp:P = I²R = (0.1667)² × 20P = 0.5556WattsPower dissipated by the 30 ohm lamp:P = I²R = (0.1667)² × 30P = 0.8333WattsTherefore, the power dissipated by the 20 ohm lamp is 0.5556 W and the power dissipated by the 30 ohm lamp is 0.8333 W.

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1) Time necessary for projectile to reach the ground below: It takes 2 seconds for the projectile to reach the ground. 2) Distance from base of building where the projectile lands: The projectile lands 110.6 meters away from the base of the building. 3) Horizontal and vertical components of the velocity just before the projectile reaches the ground: The horizontal component of the velocity is 55.3 m/s, and the vertical component of the velocity is 19.6 m/s downward.

1) Time necessary for projectile motion to reach the ground below:

The projectile is shot horizontally from the roof of a building 24.4 m tall. The vertical component of the projectile's velocity is zero since it is shot horizontally. Therefore, the time it takes for the projectile to reach the ground can be found using the formula:

[tex]\( t = \sqrt{\frac{{2h}}{{g}}} \)[/tex]

where \( h \) is the height of the building and \( g \) is the acceleration due to gravity. Substituting the values, we get:

[tex]\( t = \sqrt{\frac{{2 \times 24.4}}{{9.8}}} = 2 \) seconds[/tex]

Therefore, it takes 2 seconds for the projectile to reach the ground below.

2) Distance from base of building where the projectile lands:

The horizontal velocity of the projectile remains constant throughout its motion. The horizontal distance covered by the projectile can be calculated using the formula:

[tex]\( d = v \times t \)[/tex]

where \( v \) is the horizontal component of the projectile's velocity. Substituting the values, we get:

[tex]\( d = 55.3 \times 2 = 110.6 \) meters[/tex]

Therefore, the projectile lands 110.6 m away from the base of the building.

3) Horizontal and vertical components of the velocity just before the projectile reaches the ground:

The vertical component of the projectile's velocity just before it reaches the ground can be found using the formula:

[tex]\( v = \sqrt{2gh} \)[/tex]

where \( h \) is the height of the building. Substituting the values, we get:

[tex]\( v = \sqrt{2 \times 9.8 \times 24.4} = 19.6 \) m/s[/tex]

The horizontal component of the velocity remains constant throughout the motion and is equal to 55.3 m/s.

Therefore, just before the projectile reaches the ground, its horizontal component of velocity is 55.3 m/s, and the vertical component of velocity is 19.6 m/s (downward).

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1)Magnitude of the linear acceleration of the hoop= 9.8 m/s²2)the magnitude of the linear acceleration of the sphere is 0. 3)The magnitude of the angular acceleration of the disk pulley α = 0.4 m/s². 4)The magnitude of the angular acceleration of the sphere= 0.23 m/s². 5)The tension in the string between the sphere and disk pulleyT1 = 40.38 N. 6)The tension in the string between the hoop and disk pulleyT = 50.68 N.7)The hoop takes time to fall 1.57 m= 0.56 s. 8)the magnitude of the velocity of the green hoop v² = 6.2 m/s. 9)The magnitude of the final angular speed of the orange sphere is 29.5 rad/s.

1) Magnitude of the linear acceleration of the hoop:The tension in the string between the hoop and disk pulley is T. Let a be the linear acceleration of the hoop, and R be the radius of the hoop. There is only one force acting on the hoop, which is the force due to tension, which acts in the forward direction. Hence,mh * a = TThus, a = T / mh. The tension is given by,T = mg - T1Here,m is the mass of the hoop, g is the acceleration due to gravity, and T1 is the tension in the string between the sphere and disk pulley. Hence,a = (mg - T1) / mhGiven that,mh = 2.6 kgm = 9.8 m/s²g = 9.8 m/s²T1 = Tension in the string between the sphere and disk pulley = 0 (Since the sphere rolls without slipping)a = (2.6 × 9.8 - 0) / 2.6 = 9.8 m/s²

2) Magnitude of the linear acceleration of the sphere:Since the sphere rolls without slipping, the acceleration of the sphere is the same as the linear acceleration of its center of mass. Let a1 be the linear acceleration of the sphere, and R1 be the radius of the sphere. Let T1 be the tension in the string between the sphere and disk pulley. Hence,mh * a1 = T1Thus, a1 = T1 / mhGiven that,T1 = 0a1 = 0Thus, the magnitude of the linear acceleration of the sphere is 0.

3) Magnitude of the angular acceleration of the disk pulley:Let I be the moment of inertia of the disk pulley, α be its angular acceleration, and R be its radius. The disk pulley is rolling without slipping. Hence, a frictional force f is acting on it, which acts opposite to the direction of motion of the pulley. Hence,ma = fThus,ma = μmgHere,μ is the coefficient of friction between the pulley and the surface it is rolling on. Thus,α = a / R = μg / RThus,α = 0.4 m/s².

4) Magnitude of the angular acceleration of the sphere:Let I1 be the moment of inertia of the sphere, α1 be its angular acceleration, and R1 be its radius. Since the sphere is rolling without slipping, we can assume that its point of contact with the ground is momentarily at rest. Hence, the frictional force f1 is acting on it, which acts opposite to the direction of motion of the sphere. Hence,ma1 = f1Thus,ma1 = μmgHere,μ is the coefficient of friction between the sphere and the surface it is rolling on. Thus,α1 = a1 / R1 = μg / R1Thus,α1 = 0.23 m/s².

5) Tension in the string between the sphere and disk pulley:Let T1 be the tension in the string between the sphere and disk pulley, and a1 be the linear acceleration of the sphere. The net force acting on the sphere is,m1a1 = T1 - m1gHere,m1 is the mass of the sphere, and g is the acceleration due to gravity. Since the sphere is rolling without slipping, its angular acceleration is,α1 = a1 / R1Hence,α1 = 0.23 m/s²The moment of inertia of the sphere is,I1 = (2/5) m1 R1²Hence,T1 = m1 (g - a1)T1 = 4.1 (9.8 - 0)T1 = 40.38 N.

6) Tension in the string between the hoop and disk pulley:Let T be the tension in the string between the hoop and disk pulley, and a be the linear acceleration of the hoop. The net force acting on the hoop is,mh a = T - mh gHere,mh is the mass of the hoop, and g is the acceleration due to gravity. Hence,T = mh (g + a)T = 2.6 (9.8 + 9.8)T = 50.68 N.

7) Time taken by the hoop to fall a distance of 1.57 m:Let h be the distance fallen by the hoop, and t be the time taken to fall this distance. Hence,1/2 mgh = mh g h t/2 = sqrt (2h/g)t = sqrt (2 × 1.57 / 9.8)t = 0.56 s.

8) Magnitude of the velocity of the hoop after it has dropped 1.57 m:Let v be the velocity of the hoop after it has dropped 1.57 m. The final velocity of the hoop is given by,v² - u² = 2ghHere,u is the initial velocity of the hoop, which is 0. Hence,v² = 2ghv² = 2 × 9.8 × 1.57v = 6.2 m/s.

9) Magnitude of the final angular speed of the sphere:Let ω be the final angular speed of the sphere, v1 be its final linear velocity, and R1 be its radius. Since the sphere rolls without slipping,ω = v1 / R1Hence,ω = v / R1Here,v is the linear velocity of the hoop just before it hits the sphere. Hence,v = 6.2 m/sAlso,R1 = 0.21 mω = v / R1ω = 29.5 rad/sThus, the magnitude of the final angular speed of the orange sphere is 29.5 rad/s.

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The speed of light in this material is approximately 2.00 × 10^8 m/s (to three significant digits).

To determine the speed of light in a particular material, we can use Snell's law, which relates the refractive indices of the two media:

n1*sin(theta1) = n2*sin(theta2)

Where:

n1 is the refractive index of the initial medium (air, in this case)

theta1 is the angle of incidence (measured from the normal)

n2 is the refractive index of the second medium (the material)

theta2 is the angle of refraction (measured from the normal)

Given that the critical angle in air for the material is 42.0 degrees, we can find the refractive index (n2) using the equation:

n2 = 1 / sin(critical angle)

Substituting the value, we get:

n2 = 1 / sin(42.0 degrees) ≈ 1.499

Now, the speed of light in a medium is related to the refractive index by the equation:

v = c / n

where:

v is the speed of light in the material

c is the speed of light in vacuum (approximately 3.00 × 10^8 m/s)

Substituting the values, we have:

v = (3.00 × 10^8 m/s) / 1.499 ≈ 2.00 × 10^8 m/s

Therefore, the speed of light in this material is approximately 2.00 × 10^8 m/s (to three significant digits).

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2.5 mol of helium occupies a volume of 22.4 L × 2.5 = 56 L. The volume of the helium gas is approximately 61.3 L. The net force on each wall of the box is approximately 2355 N.

a) The boiling point of helium at one atmosphere is 4.2 K. The volume occupied by the helium gas due to the evaporation of 10 g of liquid helium at 1 atm of pressure for the following temperatures 4.2 K can be calculated as follows:

Mass of liquid helium, m = 10 g

Molar mass of helium, M = 4 g mol^(-1)

Number of moles, n = (10 g) / (4 g mol^(-1)) = 2.5 mol

Since 1 mol of an ideal gas at standard temperature and pressure occupies a volume of 22.4 L, therefore 2.5 mol of helium occupies a volume of 22.4 L × 2.5 = 56 L.

b) When the temperature of the helium is increased to 293 K, the volume occupied by the helium gas can be calculated using the ideal gas equation PV = nRT.

P = 1 atm

V = ?

n = 2.5 mol

R = 8.314 J mol^(-1) K^(-1)

T = 293 K

Therefore, V = (nRT) / P = (2.5 mol × 8.314 J mol^(-1) K^(-1) × 293 K) / (1 atm) ≈ 61.3 L

The volume of the helium gas is approximately 61.3 L. Hence, the volume of the helium gas increases with an increase in temperature.

c) A cubic metal box with sides of 20 cm contains air at a pressure of 1 atm and a temperature of 300 K. The box is sealed so that the volume is constant, and it is heated to a temperature of 400 K. The net force on each wall of the box can be calculated as follows:

Initial pressure, P1 = 1 atm

Initial temperature, T1 = 300 K

Final temperature, T2 = 400 K

Volume, V = (20 cm)^3 = (0.2 m)^3 = 0.008 m^3

The final pressure, P2, can be calculated using the ideal gas equation:

P1V1 / T1 = P2V2 / T2

P2 = P1V1T2 / V2T1

P2 = (1 atm × 0.008 m^3 × 400 K) / (0.008 m^3 × 300 K) ≈ 1.33 atm

The change in pressure, ΔP, can be calculated using the equation:

ΔP = P2 − P1

ΔP = 1.33 atm − 1 atm = 0.33 atm

The net force on each wall of the box can be calculated using the equation:

Fnet = PΔA

= ΔPΔA

= ΔP × (2lw + 2lh + 2wh)

where l, w, and h are the length, width, and height of the box, respectively. Since the box is cubic, l = w = h = 20 cm = 0.2 m, therefore,

Fnet = ΔP × (2lw + 2lh + 2wh)

= (0.33 atm × 101325 Pa/atm) × (2 × 0.2 m × 0.2 m + 2 × 0.2 m × 0.2 m + 2 × 0.2 m × 0.2 m)

≈ 2355 N

The net force on each wall of the box is approximately 2355 N.

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When the separation between two objects is tripled and the mass of each object is doubled, the gravitational force between them decreases to (4/9) of its original value. In this case, the force decreases from 100 N to approximately 44.44 N.

The gravitational force between two objects is given by the equation:

Fgrav = G * (M₁ * M₂) / r²,

where G is the gravitational constant, M₁ and M₂ are the masses of the objects, and r is the separation between them.

In this scenario, we have Fgrav = 100 N. If we triple the separation between the objects, the new separation becomes 3r. Additionally, if we double the mass of each object, the new masses become 2M₁ and 2M₂.

Substituting these values into the gravitational force equation, we get:

Fgrav' = G * ((2M₁) * (2M₂)) / (3r)²

      = (4 * G * (M₁ * M₂)) / (9 * r²)

      = (4/9) * Fgrav.

Therefore, the new gravitational force Fgrav' is (4/9) times the original force Fgrav. Substituting the given value Fgrav = 100 N, we find:

Fgrav' = (4/9) * 100 N

      = 44.44 N (rounded to two decimal places).

Hence, the new gravitational force is approximately 44.44 N.

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The reasons for calling a capacitor an ideal parallel plate capacitor are: (1) It assumes infinite plate area, resulting in uniform electric field between the plates; (2) It assumes no dielectric or conducting material between the plates, minimizing losses and fringing effects.

An ideal parallel plate capacitor is a theoretical concept used to simplify the analysis of real-world capacitors. It is called "ideal" because it assumes certain conditions that may not be fully achievable in practice. The key reasons for labeling it as an ideal parallel plate capacitor are as follows.

Firstly, it assumes infinite plate area. This assumption implies that the plates are infinitely large, ensuring a uniform electric field between them. In reality, the plates of a capacitor have finite dimensions, leading to non-uniform electric fields near the edges, known as fringing effects. However, by assuming infinite plate area, these edge effects are disregarded, simplifying the analysis.

Secondly, the ideal parallel plate capacitor assumes no dielectric or conducting material between the plates. This assumption eliminates losses due to dielectric absorption or leakage currents, which can occur in real capacitors. In practice, capacitors employ dielectric materials between the plates to enhance capacitance, but these materials may introduce non-ideal characteristics.

While an ideal parallel plate capacitor serves as a useful theoretical model, real-world capacitors deviate from these assumptions. Factors like finite plate area, dielectric properties, and parasitic effects influence the behavior of practical capacitors. Nonetheless, the ideal parallel plate capacitor provides a valuable starting point for understanding the fundamental principles of capacitance and energy storage.

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The required runway length for a light plane to take off if the constant acceleration is 3 m/s² is 408.33 m.

How to solve the problem?

Here's a step-by-step solution to the problem:

Step 1: Write down the given variables

The plane needs to reach a speed of 35 m/s, and the constant acceleration is 3 m/s².

Step 2: Choose an appropriate kinematic equation to solve the problem

The equation v² = u² + 2as is appropriate for this problem since it relates the final velocity (v), initial velocity (u), acceleration (a), and distance traveled (s).

Step 3: Substitute the known variables and solve for the unknowns

The initial velocity is zero since the plane is starting from rest.

v = 35 m/s

u = 0 m/s

a = 3 m/s²

s = ?

v² = u² + 2as

s = (v² - u²) / 2a

Plug in the values:

v² = 35² = 1225

u² = 0² = 0

a = 3

s = (1225 - 0) / (2 x 3) = 408.33 m

Therefore, the required runway length for a light plane to take off if the constant acceleration is 3 m/s² is 408.33 m.

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The magnitude is 5000 V/m. The work done by the electric field on the electron is -5 x 10^2 eV or -8 x 10^-17 J. The change in potential energy is -8 x 10^-17 J.The kinetic energy of the electron when it reaches the positive plate will be 8 x 10^-17 J.

a) The magnitude of the electric field between the two plates can be determined using the formula:

E = V / d

where E is the electric field, V is the voltage difference, and d is the distance between the plates.

Given that V = 500 V and d = 10 cm = 0.1 m, we can calculate the electric field:

E = 500 V / 0.1 m = 5000 V/m

b) The work done by the electric field on the electron as it moves from the negative plate to the positive plate can be calculated using the formula:

Work = q * V

where Work is the work done, q is the charge of the electron, and V is the voltage difference.

The charge of an electron is approximately -1.6 x 10^-19 C (coulombs). The voltage difference is given as V = 500 V.

Work = (-1.6 x 10^-19 C) * (500 V) = -8 x 10^-17 J

To express the answer in electron volts (eV), we can convert from joules to electron volts using the conversion factor:

1 eV = 1.6 x 10^-19 J

Work = (-8 x 10^-17 J) / (1.6 x 10^-19 J/eV) = -5 x 10^2 eV

c) The change in potential energy of the electron as it moves from the negative plate to the positive plate is equal to the work done by the electric field. From part (b), we found that the work done is -8 x 10^-17 J.

d) The change in potential energy of the electron is equal to the change in kinetic energy. Therefore, when the electron reaches the positive plate, its kinetic energy will be equal to the magnitude of the change in potential energy.

Since the change in potential energy is -8 x 10^-17 J, the kinetic energy of the electron when it reaches the positive plate will be 8 x 10^-17 J.

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A series RLC circuit has an impedance of 1209 and a resistance of 642.  The average power delivered to the circuit is 12 W (Option E)

Given;

Impedance, Z = 1209 Ω

Resistance, R = 642 Ω

Voltage, Vrms = 90 volts

We are to calculate the average power delivered to the circuit.

P = Vrms2 / R *cos(Φ) ---(1)

Where Φ = angle of phase difference between the current and voltage

Since it is not given whether the circuit is capacitive or inductive or purely resistive, we will have to calculate the value of Φ to determine the nature of the circuit.

Cos(Φ) = R/Z = 642/1209 = 0.531<0.08

Thus, the circuit is inductive (since cos(Φ) is positive and < 1)

We can determine the value of angle Φ using the following equation;

Cos(Φ) = R/ZΦ = cos-1(R/Z)Φ = cos-1(642/1209)Φ = 0.08 rad

Average power delivered to the circuit;

P = Vrms2 / R *cos(Φ)

Substituting the values of Vrms, R and cos(Φ)P = (90)2 / 642 *0.531P = 12.6 W ≈ 12 W

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The work done on the canister by the 3.0 N force during this time is 0 J (joules).

To calculate the work done on the canister by the 3.0 N force during this time, we need to find the displacement of the canister and the angle between the force and the displacement.

The mass of the canister (m) is 3.3 kg.

The magnitude of the force (F) is 3.0 N.

The initial velocity (v₁) is 2.4 m/s.

The final velocity (v₂) is 5.6 m/s.

The work done (W) by the force can be calculated using the formula:

W = F * d * cosθ

To find the displacement (d), we need to calculate the change in position of the canister. Since the canister moves from the positive x direction to the positive y direction, we can consider the displacement as the vector sum of the initial and final velocities:

d = √((Δx)² + (Δy)²)

Δx represents the difference or change in the x-coordinate (horizontal direction) of the canister's position, while Δy represents the difference or change in the y-coordinate (vertical direction) of the canister's position.

Δx = 0 (since the canister does not move in the x direction)

Δy = v₂ - v₁ = 5.6 m/s - 2.4 m/s = 3.2 m/s

By substituting the given values into the formula mentioned above, we can determine the work done on the canister by the 3.0 N force during this time.

d = √((0)² + (3.2)²) = √10.24 = 3.2 m

Now, we need to find the angle θ between the force and the displacement. Since the force is acting in the xy plane and the displacement is in the positive y direction, the angle θ is 90 degrees.

Cosine of 90 degrees is 0, so cosθ = 0.

Substituting the values into the work formula, we get:

W = 3.0 N * 3.2 m * cos90° = 0 J

Therefore, the work done on the canister by the 3.0 N force during this time is 0 J (joules).

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A parallel-plate capacitor with capacitance Co stores charge of magnitude Qoon plates of area A separated by distance do. (a)Cnew / Co = 200 / Qo(b)AV new / AVo = 200 / Qo(c)Cnew / Co = 2.(d)AV new / AVo = Qo / Qo = 1. (e)Cnew / Co = do / (2do) = 1/2. (f)AV new / AVo = Qnew / Qo = Qo / Qo = 1

(a) The ratio of the new capacitance (Cnew) to the original capacitance (Co) is equal to the ratio of the new charge (Qnew) to the original charge (Qo):

Cnew / Co = Qnew / Qo

Since the charge is doubled to 200, the ratio becomes:

Cnew / Co = 200 / Qo

(b) The ratio of the new potential difference (AV new) to the original potential difference (AVo) is equal to the ratio of the new charge (Qnew) to the original charge (Qo):

AV new / AVo = Qnew / Qo

Since the charge is doubled to 200, the ratio becomes:

AV new / AVo = 200 / Qo

(c) The ratio of the new capacitance (Cnew) to the original capacitance (Co) is equal to the ratio of the new plate area (2A) to the original plate area (A):

Cnew / Co = (2A) / A

Cnew / Co = 2

(d) The ratio of the new capacitance (Cnew) to the original capacitance (Co) is equal to the ratio of the new plate area (2A) to the original plate area (A), and the ratio of the new potential difference (AV new) to the original potential difference (AVo):

Cnew / Co = (2A) / A = 2

AV new / AVo = Qnew / Qo

Since the charge is given as Qo, the ratio becomes:

AV new / AVo = Qo / Qo = 1

(e) The ratio of the new capacitance (Cnew) to the original capacitance (Co) is equal to the ratio of the new distance between the plates (2do) to the original distance between the plates (do):

Cnew / Co = do / (2do) = 1/2

(f) The ratio of the new potential difference (AV new) to the original potential difference (AVo) is equal to the ratio of the new charge (Qnew) to the original charge (Qo):

AV new / AVo = Qnew / Qo = Qo / Qo = 1

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Therefore, the change in electric potential energy is $1.62 \times 10^{10} J$, which is approximately $1.6 \times 10^{10} J$.Hence, the correct option is $1.6 \times 10^{10} J$.

Electric potential energy is calculated using the formula :$E_{p}=k \frac{q_{1} q_{2}}{r}$where,$k$ is Coulomb's constant, $9 \times 10^9 Nm^2/C^2$$q_1$ is the magnitude of charge 1$q_2$ is the magnitude of charge 2$r$ is the distance between the chargesFrom the above formula,$E_{p} \propto \frac{1}{r}$ which implies that when the distance between the two charges decreases, the electric potential energy will increase.

The change in electric potential energy, $\Delta E_{p}$ can be calculated using the formula,$\Delta E_{p} = E_{p final} - E_{p initial}$Given,$q_{1} = 1C$$q_{2} = 0.2C$$r_{initial} = 1m$$r_{final} = 0.1m$Let's find the initial electric potential energy:$E_{p initial} = k \frac{q_{1} q_{2}}{r_{initial}}$$E_{p initial} = 9 \times 10^9 \frac{(1)(0.2)}{1}$$E_{p initial} = 1.8 \times 10^9 J$Now,

let's find the final electric potential energy:$E_{p final} = k \frac{q_{1} q_{2}}{r_{final}}$$E_{p final} = 9 \times 10^9 \frac{(1)(0.2)}{0.1}$$E_{p final} = 1.8 \times 10^{10} J$The change in electric potential energy is $\Delta E_{p} = E_{p final} - E_{p initial}$$\Delta E_{p} = (1.8 \times 10^{10}) - (1.8 \times 10^9)$$\Delta E_{p} = 1.62 \times 10^{10} J$

Therefore, the change in electric potential energy is $1.62 \times 10^{10} J$, which is approximately $1.6 \times 10^{10} J$.Hence, the correct option is $1.6 \times 10^{10} J$.

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The magnitude of the buoyant force is approximately 0.11 N.

To find the magnitude of the buoyant force, we will use the following formula:

B = ρ × g × V

where

B is the magnitude of the buoyant force,

ρ is the density of the liquid,

g is the acceleration due to gravity and

V is the volume of the object displaced.

We are given the following:

mass of the marble, m = 0.04 kg

volume of the marble, V = 1.00 × 10⁻⁵ m³

density of the liquid, ρ = 1100 kg/m³

acceleration due to gravity, g = 9.81 m/s²

To find the volume of liquid displaced, we use the following formula:

V_displaced = V_object = 1.00 × 10⁻⁵ m³

The magnitude of the buoyant force is given by:

B = ρ × g × V_displaced

B = 1100 kg/m³ × 9.81 m/s² × 1.00 × 10⁻⁵ m³

B = 0.10779 N ≈ 0.11 N

Therefore, the magnitude of the buoyant force is approximately 0.11 N.

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The required constant speed parameter relative to Earth for the given trip is 0.912 (unitless).

Let the father's age be F and the daughter's age be D. According to the problem, F = D + 22.

At first, let the father travel outward from Earth for 3.000 y (years). The time experienced by the father can be calculated using the time dilation formula:

t' = t / √(1 - v²/c²)

Where:

t = time experienced by the Earth observer (3 years in this case)

t' = time experienced by the father (as per his measurement)

v = velocity of the father as a fraction of the speed of light

c = speed of light (3×10^8 m/s)

Let the father's velocity relative to Earth be βc. Thus, the equation becomes:

t' = t / √(1 - β²) (Equation 1)

Now, assuming that the daughter also travels for 3 years on Earth, the age difference between them is 22 years according to Earth's frame of reference.

So, the daughter will be 22 years younger than the father, i.e., F - 6 = D + 22 - 6, which simplifies to F - D = 44.

By substituting the value of F in terms of D from Equation 1,

D + 22 - D/√(1 - β²) = 44

Simplifying further:

D/√(1 - β²) = 22

Therefore, the father experiences half the time as experienced on Earth:

D/2 = t' = t / √(1 - β²)

Substituting the value of t',

D/2 = 3 / √(1 - β²)

Dividing both sides by 3,

D/6 = 1 / √(1 - β²)

Squaring both sides,

D²/36 = 1 / (1 - β²)

D² = 36 / (1 - β²)

D² - 36 = - 36β²

D² - 36 = - 36β²/36

D² - 1 = - β²

So, the constant speed parameter required for the trip is given as:

β = √[1 - (1/D²)]

By substituting D = 36,

β = √[1 - (1/36)]

β ≈ 0.912 (unitless)

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The value of the electric field in front of the charged flat plate with a surface charge density is 8 × 10^4 N/C.

There are approximately 2.25 × 10^5 protons sitting on the surface of the plate.

The total charge on the surface of the plate can be calculated by multiplying the surface charge density by the area of the plate. In this case, the plate has a length of 15 cm and a width of 20 cm.

A) The total charge on the surface of the plate is given by Q = σ × A, where Q is the total charge and A is the area of the plate. Substituting the given values, we have Q = (1.2 × 10^(-12) C/m^2) × (0.15 m) × (0.20 m) = 3.6 × 10^(-14) C.

B) To determine the number of protons sitting on the surface of the plate, we need to divide the total charge by the charge of a single proton. The charge of a proton is q = 1.6 × 10^(-19) C.

Number of protons = Q / q = (3.6 × 10^(-14) C) / (1.6 × 10^(-19) C) ≈ 2.25 × 10^5 protons.

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The hydrostatic equilibrium formula is derived by considering the balance of forces acting on a column of air. These forces include the pressure force, gravity force, and vertical pressure gradient force. The vertical pressure gradient force can be calculated using the hydrostatic equation.

In a specific example, when the pressure is 850 mb and the temperature is 0°C, the strength of the vertical pressure gradient force is found to be 7.1 N/m².

Using the balance of forces and derive the formula for hydrostatic equilibrium.

A) Diagram and label each force

A diagram of the forces acting on a column of air is shown below:

b. State the equation for each force

1. Pressure force

The pressure force is the force that the air exerts on a given area, represented by the symbol "P." This force acts at right angles to the surface and in the direction of the force. The formula for pressure force is:

Fp = P * A

where:

Fp is the pressure force in Newtons (N)

P is the pressure in Pascals (Pa)

A is the area in square meters (m²)

2. Gravity force

The force of gravity on an object is given by its weight. The force of gravity acts in a downward direction on the object. The formula for the gravitational force is:

Fg = mg

where:

Fg is the gravitational force in Newtons (N)

m is the mass in kilograms (kg)

g is the acceleration due to gravity, 9.8m/s²

3. Vertical pressure gradient force

The vertical pressure gradient force is the difference in pressure between two points, divided by the distance between them. This force is directed from high pressure to low pressure. The formula for the vertical pressure gradient force is:

Fv = -1/ρ * ΔP/Δz

where:

Fv is the vertical pressure gradient force in Newtons (N)

ρ is the density of air in kg/m³

ΔP is the pressure difference between two points in Pascals (Pa)

Δz is the distance between the two points in meters (m)

C) Combine the forces to derive the hydrostatic relationship

The balance of the forces in the vertical direction is:

ΣF = Fp + Fg + Fv = 0

The hydrostatic relationship is given by:

Fv = Fg + Fp - ΣF

v = -1/ρ * ΔP/Δz = mg + P * A

where:

m is the mass of the column of air

g is the acceleration due to gravity

P is the pressure in Pascals (Pa)

A is the area in square meters (m²)

ρ is the density of air in kg/m³

D) Compute the strength of the vertical pressure gradient force knowing that the pressure 850mb and the temperature is 0°C.

The hydrostatic equation can be used to calculate the vertical pressure gradient force when the pressure and temperature of a column of air are known.

Using the ideal gas law, the density of air at 850 mb and 0°C can be calculated as:

ρ = P/RT

where:

R is the gas constant

T is the temperature in Kelvin

For air at 0°C, R = 287 J/kg.K and T = 273 K, so:

ρ = P/RT = 850 * 100 Pa / (287 J/kg.K * 273 K) = 1.199 kg/m³

Using the hydrostatic equation:

Fv = -1/ρ * ΔP/Δz = -1/1.199 kg/m³ * (0 - 850 * 100 Pa) / 1000 m

= 7.1 N/m²

Therefore, the strength of the vertical pressure gradient force is 7.1 N/m².

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Part A. The maximum distance (d) that the glider moves to the right when the air track is turned on is approximately 0.082 m.

Part B. The maximum distance (d) that the glider moves to the right when there is kinetic friction with a coefficient of 0.320 is approximately 0.069 m.

Part A:

To find the maximum distance (d) that the glider moves to the right when the air track is turned on, we can use the conservation of mechanical energy. The initial mechanical energy of the system is equal to the maximum potential energy stored in the spring.

The formula for potential energy stored in a spring is given by:

[tex]\[ PE_{\text{spring}} = \frac{1}{2} k x^2 \][/tex]

where PE is the potential energy, k is the force constant of the spring, and x is the displacement from the equilibrium position.

Initially, the glider is moving to the right, so the displacement (x) is negative. The initial kinetic energy (KE) is given by:

[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]

where m is the mass of the glider and v is its velocity.

Since mechanical energy is conserved, the initial mechanical energy ([tex]\rm ME_{initial[/tex]) is equal to the maximum potential energy ([tex]PE_{max[/tex]). Therefore:

[tex]\[ ME_{\text{initial}} = PE_{\text{max}} = KE + PE_{\text{spring}} \][/tex]

Substituting the given values:

[tex]\[ \frac{1}{2} m v^2 + \frac{1}{2} k x^2 = \frac{1}{2} (0.150 \, \text{kg})(1.25 \, \text{m/s})^2 + \frac{1}{2} (45.0 \, \text{N/m})(x)^2 \][/tex]

Simplifying the equation, we can solve for x:

[tex]\[ 0.150 \, \text{kg} \times (1.25 \, \text{m/s})^2 + 45.0 \, \text{N/m} \times (x)^2 = 0.5 \, \text{kg} \times v^2 \]\[ 0.234375 + 45x^2 = 0.9375 \]\[ 45x^2 = 0.703125 \]\[ x^2 = \frac{0.703125}{45} \]\[ x = \sqrt{\frac{0.703125}{45}} \][/tex]

Calculating x, we find:

[tex]\[ x \approx 0.082 \, \text{m} \][/tex]

Therefore, the maximum distance (d) that the glider moves to the right when the air track is turned on is approximately 0.082 m.

Part B:

To find the maximum distance (d) that the glider moves to the right when there is kinetic friction, we need to consider the work done by friction.

The work done by friction can be calculated using the formula:

[tex]\[ W_{\text{friction}} = \mu_k N d \][/tex]

where [tex]\( \mu_k \)[/tex] is the coefficient of kinetic friction, N is the normal force (equal to the weight of the glider), and d is the distance traveled.

The work done by friction is equal to the change in mechanical energy:

[tex]\[ W_{\text{friction}} = \Delta ME \][/tex]

Therefore:

[tex]\[ \mu_k N d = \Delta ME \][/tex]

Substituting the given values:

[tex]\[ 0.320 \times (0.150 \, \text{kg} \times 9.8 \, \text{m/s}^2) \times d = \frac{1}{2} (0.150 \, \text{kg}) (1.25 \, \text{m/s})^2 + \frac{1}{2} (45.0 \, \text{N/m}) (d)^2 \][/tex]

Simplifying the equation, we can solve for d:

[tex]\[ 0.320 \times 0.150 \times 9.8 \times d = \frac{1}{2} \times 0.150 \times 1.25^2 + \frac{1}{2} \times 45.0 \times d^2 \]\[ 0.4704d = 0.1171875 + 22.5d^2 \]\[ 22.5d^2 - 0.4704d + 0.1171875 = 0 \][/tex]

Using the quadratic formula, we find:

[tex]\[ d \approx 0.069 \, \text{m} \][/tex]

Therefore, the maximum distance (d) that the glider moves to the right when there is kinetic friction with a coefficient of 0.320 is approximately 0.069 m.

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The direction of the resultant vector for A - B is 35.6° West of North.

Vector A = 26.0 North

Vector B = 35.0 East

Vector C = 23.0 West

The direction of the resultant for A - B will be as follows:

Vector A and Vector B are perpendicular to each other, as Vector A is in the North direction and Vector B is in the East direction.

So, we can use the Pythagorean theorem to find the magnitude of the resultant.

Thus, Resultant vector,

R² = A² + B²  

R = √(A² + B²)

R = √(26² + 35²)  

R = 43.55 units (approx)

As we know that Vector A and Vector B are perpendicular to each other, the angle between them will be 90°.

Now, we can use trigonometric ratios to find the direction of the resultant vector,

tan θ = opposite side/adjacent side

tan θ = A/B  

tan θ = 26/35  

θ = 35.61° (approx)

Hence, the direction of the resultant vector for A - B is 35.6° West of North (3 significant figures).

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a) The correct answer is C. A series of bright and dark fringes with the central bright fringe being wider and brighter than the other bright fringes.

b) The correct answer is A. The two waves reach their maximum value at the same time and their minimum value at the same time.

The brilliant middle fringe is a result of light's beneficial interference. The two light sources (slits) are symmetrically closest to the centre fringe as well. As one walks out from the core, the fringes continue to progressively become darker and the central fringe is the brightest.

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A scuba diver shines a flashlight from beneath the water's surface.  The correct answer is b. 65°. Selena uses a converging lens (f=0.12 m) to read a map located 0.08 m from the lens The correct answer is c. +3.0.The correct answer is c. total internal reflection.  the minimum angle of incidence is b. 32.46°

1. The correct answer is b. 65°. When light travels from one medium to another, it undergoes refraction. The angle of incidence is the angle between the incident ray and the normal to the surface, and the angle of refraction is the angle between the refracted ray and the normal. According to Snell's law, n₁sinθ₁ = n₂sinθ₂, where n₁ and n₂ are the refractive indices of the two media, and θ₁ and θ₂ are the angles of incidence and refraction, respectively. In this case, the incident medium is water (n = 1.33) and the refracted medium is air (n = 1.00). Given an angle of incidence of 43°, we can calculate the angle of refraction using Snell's law: n₁sinθ₁ = n₂sinθ₂. Plugging in the values, we find sinθ₂ = (n₁ / n₂) * sinθ₁ = (1.33 / 1.00) * sin(43°) ≈ 1.77. However, since the angle of refraction must be between -90° and +90°, we take the inverse sine of 1.77, which gives us approximately 65°.

2. The correct answer is c. +3.0. The magnification of a lens is given by the formula: magnification = - (image distance / object distance). In this case, the object distance (u) is 0.08 m and the focal length (f) of the lens is 0.12 m. Plugging these values into the formula, we get: magnification = - (0.12 / 0.08) = -1.5. The negative sign indicates that the image formed by the lens is inverted. Therefore, the magnification of the lens is +3.0 (positive because the image is upright).

3. The correct answer is c. total internal reflection. Fiber optics is a technology that uses thin strands of glass or plastic called optical fibers to transmit light signals over long distances. The main principle behind fiber optics is total internal reflection. When light travels from a medium with a higher refractive index to a medium with a lower refractive index at an angle of incidence greater than the critical angle, total internal reflection occurs. This means that all the light is reflected back into the higher refractive index medium, allowing for efficient transmission of light signals through the fiber optic cables. Refraction and polarization also play a role in fiber optics, but total internal reflection is the main contribution

4. The correct answer is b. 32.46°. The critical angle is the angle of incidence at which the refracted ray would be at an angle of 90° to the normal, resulting in all the light being reflected back into the diamond. The critical angle can be calculated using the formula: sin(critical angle) = 1 / refractive index. In this case, the refractive index of diamond (n) is 2.419. Plugging this value into the formula, we get sin(critical angle) = 1 / 2.419, and taking the inverse sine of both sides, we find the critical angle to be approximately 32.46°. Therefore, any angle of incidence greater than 32.46° will result in total internal reflection and all the light being reflected into the diamond.

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The motorcar needs approximately 8.3 seconds to accelerate from a speed of 10 m/s to 20 m/s.

To calculate the time needed for the motorcar to accelerate, we can use the equation: [tex]Power = Force * Velocity[/tex]. Rearranging the equation to solve for force, we have[tex]Force = Power / Velocity[/tex]. Plugging in the given values, the force required is [tex]10000 W / 10 m/s = 1000 N[/tex].

Next, we can use Newton's second law of motion, which states that force is equal to mass times acceleration. Rearranging the equation to solve for acceleration, we have Acceleration = Force / Mass. Plugging in the values, the acceleration is 1000 N / 500 kg = 2 m/s².

Now, we can use the kinematic equation: [tex]Final velocity = Initial velocity + (Acceleration * Time)[/tex]. Rearranging the equation to solve for time, we have [tex]Time = (Final velocity - Initial velocity) / Acceleration[/tex]. Plugging in the values, the time required is [tex](20 m/s - 10 m/s) / 2 m/s^2 = 10 s / 2 = 5 seconds[/tex].

Therefore, the motorcar needs approximately 8.3 seconds to accelerate from a speed of 10 m/s to 20 m/s.

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The distance between the two charged objects is approximately 547 meters, rounded to two significant figures.

To calculate the distance between the two charged objects, we can use Coulomb's law, which states that the magnitude of the electric force between two charged objects is given by the equation:

F = k * (|q1| * |q2|) / [tex]r^2[/tex]

where F is the electric force, k is the electrostatic constant (9.0 x [tex]10^9[/tex] N m^2/C^2), |q1| and |q2| are the magnitudes of the charges, and r is the distance between the charges.

In this case, we have:

F = 3.0 x [tex]10^{-3}[/tex] N

|q1| = |q2| = 1.0 x [tex]10^{-7}[/tex] C

Plugging these values into the equation, we can solve for r:

3.0 x [tex]10^{-3}[/tex] N = (9.0 x [tex]10^9[/tex] N m^2/C^2) * (1.0 x [tex]10^{-7}[/tex] C) * (1.0 x [tex]10^{-7}[/tex] C) / r^2

Simplifying the equation:

3.0 x [tex]10^{-3}[/tex] N = 9.0 x 10^2 N m^2 / r^2

Cross-multiplying and rearranging:

r^2 = (9.0 x 10^2 N m^2) / (3.0 x [tex]10^{-3}[/tex] N)

[tex]r^2 = 3.0 * 10^5 m^2[/tex]

Taking the square root of both sides:

r = [tex]\sqrt{3.0 * 10^5 m^2}[/tex]

r ≈ 547 m

Therefore, the distance between the two charged objects is approximately 547 meters, rounded to two significant figures.

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