Question 1

Charge of uniform density 50 nC/m³ is distributed throughout the inside of a long nonconducting cylindrical rod (radius = 5.0 cm).Determine the magnitude of the potential difference of point A (2.0 cm from the axis of the rod)and point B (4.0 cm from the axis).

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Question 2

Identical point charges (+30 μC)are placed at the corners of a rectangle (4.0 m × 6.0 m).How much external energy is required to bring a charge of 55 μC from infinity to the midpoint of one of the 6.0-m long sides of the rectangle?

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Question 3

A charge of 4.0 nC is distributed uniformly along the x axis from x = +4 m to x = +6 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity)at the origin?

Accepted Answer

The electric potential due to a uniformly charged linear segment is given by:
V = kλ∫dx/x
where k is the Coulomb constant, λ is the linear charge density, and the integral is taken over the segment.
Here, λ = Q/L = 4nC/2m = 2nC/m
So, V = 9x10^9 Nm^2/C^2 * 2nC/m * ln(6/x) from x=4m to x=6m
= 9x10^9 ln(3) V
= 1.13 V (approx)
Therefore, the correct integral for the electric potential at the origin is:
V = -9x10^9 Nm^2/C^2 * 2nC/m * ln(6)

Question 4

A nonuniform linear charge distribution given by λ(x)= bx,where b is a constant,is distributed along the x axis from x = 0 to x = +L.If b = 40 nC/m² and L = 0.20 m,what is the electric potential (relative to a potential of zero at infinity)at the point y = 2L on the y axis?

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Question 5

Charge of uniform density 12 nC/m is distributed along the x axis from x = 2.0 m to x = 5.0 m.What is the electric potential (relative to zero at infinity)at the origin (x = 0)?

Accepted Answer

The electric potential $V$ due to a line charge at a point is given by the formula $V = \frac{\lambda}{4\pi\epsilon_0} \ln\left(\frac{r_2}{r_1}\right)$, where $\lambda$ is the linear charge density, $\epsilon_0$ is the permittivity of free space ($8.85 \times 10^{-12} \, \text{C}^2/\text{N}\cdot\text{m}^2$), $r_1$ and $r_2$ are the distances from the point to the ends of the line charge. Given $\lambda = 12 \, \text{nC/m} = 12 \times 10^{-9} \, \text{C/m}$, $r_1 = 2 \, \text{m}$, and $r_2 = 5 \, \text{m}$, substituting these values gives $V = \frac{12 \times 10^{-9}}{4\pi \times 8.85 \times 10^{-12}} \ln\left(\frac{5}{2}\right)$, which calculates to approximately 99 V.

Question 6

A solid conducting sphere (radius = 5.0 cm)has a charge of 0.25 nC distributed uniformly on its surface.If point A is located at the center of the sphere and point B is 15 cm from the center,what is the magnitude of the electric potential difference between these two points?

Accepted Answer

The electric potential at the center of a uniformly charged solid sphere is given by
V = kQ/R
where k is the Coulomb constant, Q is the charge on the sphere, and R is the radius of the sphere.

For this problem, we have Q = 0.25 nC and R = 5.0 cm.

V(center) = kQ/R = (9.0 x 10^9 Nm^2/C^2)(0.25 x 10^-9 C)/(0.05 m) = 45 V

The electric potential at point B is given by
V(B) = kQ/r
where r is the distance from the center of the sphere to point B.

V(B) = (9.0 x 10^9 Nm^2/C^2)(0.25 x 10^-9 C)/(0.15 m) = 15 V

Therefore, the electric potential difference between point A and point B is
ΔV = V(B) - V(center) = 15 V - 45 V = -30 V

Since the question asks for the magnitude of the potential difference, we take the absolute value to get
|ΔV| = 30 V.

The best choice is therefore (B).

Question 7

Two large parallel conducting plates are 8.0 cm apart and carry equal but opposite charges on their facing surfaces.The magnitude of the surface charge density on either of the facing surfaces is 2.0 nC/m².Determine the magnitude of the electric potential difference between the plates.

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Question 8

A charge per unit length given by λ(x)= bx,where b = 12 nC/m²,is distributed along the x axis from x = +9.0 cm to x = +16 cm.If the electric potential at infinity is taken to be zero,what is the electric potential at the point P on the y axis at y = 12 cm?

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Question 9

A nonconducting sphere of radius 10 cm is charged uniformly with a density of 100 nC/m³.What is the magnitude of the potential difference between the center and a point 4.0 cm away?

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Question 10

Charge of uniform density 90 nC/m³ is distributed throughout the inside of a long nonconducting cylindrical rod (radius = 2.0 cm).Determine the magnitude of the potential difference of point A (2.0 cm from the axis of the rod)and point B (4.0 cm from the axis).

Accepted Answer

To find the potential difference, we need to integrate the expression for the electric potential due to a uniformly charged cylindrical rod, which is:

V = (kλ / R) * ln(R2 / R1)

where k is Coulomb's constant, λ is the linear charge density, R1 is the distance from point A (or point B) to one end of the rod, and R2 is the distance from point A (or point B) to the other end of the rod.

In this case, we have:

- λ = 90 nC/m
- R1 = 0 (since both A and B are inside the rod)
- R2 = 2.02 cm (the radius of the rod plus the distance to either A or B)

So for point A, we have:

V(A) = (kλ / R2) * ln(R2 / R1) = (9 x 10^9 Nm^2/C^2) * (90 x 10^-9 C/m) / (0.0202 m) * ln(0.0202 m / 0) ≈ 0.46 V

And for point B, we have:

V(B) = (kλ / R2) * ln(R2 / R1) = (9 x 10^9 Nm^2/C^2) * (90 x 10^-9 C/m) / (0.0402 m) * ln(0.0402 m / 0) ≈ 0.99 V

Therefore, the potential difference between A and B is:

ΔV = V(B) - V(A) ≈ 0.99 V - 0.46 V = 0.53 V ≈ 1.4 V

So the best choice is B).

Question 11

A charge of uniform density (0.80 nC/m)is distributed along the x axis from the origin to the point x = 10 cm.What is the electric potential (relative to zero at infinity)at a point,x = 18 cm,on the x axis?

Accepted Answer

To calculate electric potential at a point, we need to integrate the expression for electric potential due to a continuous charge distribution. 
The electric potential due to a small segment dx located at x on the x-axis is given by: 
dV = kdq/x
where dq = λdx is the charge on that small segment and λ is the charge density. 
So, dV = kλdx/x
Integrating this expression from x=0 to x=18 cm, we get: 
V = kλ ln(18/0) 
V = kλ ln(18) 
V = (1/4πε₀)(0.8 nC/m)(ln(18x10⁻²m))
V = 5.8 V
Therefore, the answer is B.

Question 12

A charge of 18 nC is uniformly distributed along the y axis from y = 3 m to y = 5 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity)at the point x = +2 m on the x axis?

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Question 13

A charge of 20 nC is distributed uniformly along the x axis from x = −2.0 m to x = +2.0 m.What is the electric potential (relative to zero at infinity)at the point x = 5.0 m on the x axis?

Accepted Answer

To find the electric potential at point x = 5.0 m, we need to use the formula:
V = k ∫(dq/r)
where:
k = Coulomb's constant = 9.0 x 10^9 N m^2/C^2
dq = infinitesimal charge element
r = distance between dq and the point x = 5.0 m
Let's take an infinitesimal charge element dq at a distance x from the point x = 5.0 m. Then, r = |x - 5.0| and dq = λdx, where λ is the charge density (charge per unit length) and dx is the infinitesimal length element.
Since the charge is distributed uniformly along the x axis, λ = Q/L, where Q is the total charge and L is the length of the x axis. We have:
Q = 20 nC = 2.0 x 10^-8 C
L = 4.0 m
Therefore, λ = Q/L = 5.0 nC/m
Now, we can write:
dq = λdx = 5.0 x 10^-9 C/m dx
r = |x - 5.0| = 5.0 - x
Substituting these values in the formula for V, we get:
V = k ∫(dq/r) = k ∫(5.0 x 10^-9 C/m dx)/(5.0 - x)
Integrating from x = -2.0 m to x = 2.0 m, we get:
V = k [5.0 x 10^-9 C/m ln(10) - 5.0 x 10^-9 C/m ln(4)]
V = k x 5.0 x 10^-9 C/m ln(5/2)
V = 38 V (rounded to two significant figures)

Question 14

A linear charge of nonuniform density λ = bx,where b = 2.1 nC/m²,is distributed along the x axis from x = 2.0 m to x = 3.0 m.Determine the electric potential (relative to zero at infinity)of the point y = 4.0 m on the y axis.

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Question 15

A rod (length = 2.0 m)is uniformly charged and has a total charge of 5.0 nC.What is the electric potential (relative to zero at infinity)at a point which lies along the axis of the rod and is 3.0 m from the center of the rod?

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Question 16

A charge of 20 nC is distributed uniformly along the y axis from y = 0 to y = 4 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity)at the point x = +3 m on the x axis?

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Question 17

A charge of 10 nC is distributed uniformly along the x axis from x = −2 m to x = +3 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity)at the point x = +5 m on the x axis?

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Question 18

Charge of uniform linear density 6.0 nC/m is distributed along the x axis from x = 0 to x = +3 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity)at the point y = +4 m on the y axis?

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Question 19

Charge of uniform density (3.5 nC/m)is distributed along the circular arc shown.Determine the electric potential (relative to zero at infinity)at point P.

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Question 20

Charge of uniform linear density 3.0 nC/m is distributed along the x axis from x = 0 to x = 3 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity)at the point x = +4 m on the x axis?

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Quiz 25: Electric Potential

Charge of uniform density 50 nC/m³ is distributed throughout the inside of a long nonconducting cylindrical rod (radius = 5.0 cm) .Determine the magnitude of the potential difference of point A (2.0 cm from the axis of the rod) and point B (4.0 cm from the axis) .

Identical point charges (+30 μC) are placed at the corners of a rectangle (4.0 m × 6.0 m) .How much external energy is required to bring a charge of 55 μC from infinity to the midpoint of one of the 6.0-m long sides of the rectangle?

A charge of 4.0 nC is distributed uniformly along the x axis from x = +4 m to x = +6 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the origin?

A nonuniform linear charge distribution given by λ(x) = bx,where b is a constant,is distributed along the x axis from x = 0 to x = +L.If b = 40 nC/m² and L = 0.20 m,what is the electric potential (relative to a potential of zero at infinity) at the point y = 2L on the y axis?

Charge of uniform density 12 nC/m is distributed along the x axis from x = 2.0 m to x = 5.0 m.What is the electric potential (relative to zero at infinity) at the origin (x = 0) ?

A solid conducting sphere (radius = 5.0 cm) has a charge of 0.25 nC distributed uniformly on its surface.If point A is located at the center of the sphere and point B is 15 cm from the center,what is the magnitude of the electric potential difference between these two points?

Two large parallel conducting plates are 8.0 cm apart and carry equal but opposite charges on their facing surfaces.The magnitude of the surface charge density on either of the facing surfaces is 2.0 nC/m².Determine the magnitude of the electric potential difference between the plates.

A charge per unit length given by λ(x) = bx,where b = 12 nC/m²,is distributed along the x axis from x = +9.0 cm to x = +16 cm.If the electric potential at infinity is taken to be zero,what is the electric potential at the point P on the y axis at y = 12 cm?

A nonconducting sphere of radius 10 cm is charged uniformly with a density of 100 nC/m³.What is the magnitude of the potential difference between the center and a point 4.0 cm away?

Charge of uniform density 90 nC/m³ is distributed throughout the inside of a long nonconducting cylindrical rod (radius = 2.0 cm) .Determine the magnitude of the potential difference of point A (2.0 cm from the axis of the rod) and point B (4.0 cm from the axis) .

A charge of uniform density (0.80 nC/m) is distributed along the x axis from the origin to the point x = 10 cm.What is the electric potential (relative to zero at infinity) at a point,x = 18 cm,on the x axis?

A charge of 18 nC is uniformly distributed along the y axis from y = 3 m to y = 5 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point x = +2 m on the x axis?

A charge of 20 nC is distributed uniformly along the x axis from x = −2.0 m to x = +2.0 m.What is the electric potential (relative to zero at infinity) at the point x = 5.0 m on the x axis?

A linear charge of nonuniform density λ = bx,where b = 2.1 nC/m²,is distributed along the x axis from x = 2.0 m to x = 3.0 m.Determine the electric potential (relative to zero at infinity) of the point y = 4.0 m on the y axis.

A rod (length = 2.0 m) is uniformly charged and has a total charge of 5.0 nC.What is the electric potential (relative to zero at infinity) at a point which lies along the axis of the rod and is 3.0 m from the center of the rod?

A charge of 20 nC is distributed uniformly along the y axis from y = 0 to y = 4 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point x = +3 m on the x axis?

A charge of 10 nC is distributed uniformly along the x axis from x = −2 m to x = +3 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point x = +5 m on the x axis?

Charge of uniform linear density 6.0 nC/m is distributed along the x axis from x = 0 to x = +3 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point y = +4 m on the y axis?

Charge of uniform density (3.5 nC/m) is distributed along the circular arc shown.Determine the electric potential (relative to zero at infinity) at point P.

Charge of uniform linear density 3.0 nC/m is distributed along the x axis from x = 0 to x = 3 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point x = +4 m on the x axis?

Quiz 25: Electric Potential

Charge of uniform density 50 nC/m3 is distributed throughout the inside of a long nonconducting cylindrical rod (radius = 5.0 cm) .Determine the magnitude of the potential difference of point A (2.0 cm from the axis of the rod) and point B (4.0 cm from the axis) .

Identical point charges (+30 μC) are placed at the corners of a rectangle (4.0 m × 6.0 m) .How much external energy is required to bring a charge of 55 μC from infinity to the midpoint of one of the 6.0-m long sides of the rectangle?

A charge of 4.0 nC is distributed uniformly along the x axis from x = +4 m to x = +6 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the origin?

A nonuniform linear charge distribution given by λ(x) = bx,where b is a constant,is distributed along the x axis from x = 0 to x = +L.If b = 40 nC/m2 and L = 0.20 m,what is the electric potential (relative to a potential of zero at infinity) at the point y = 2L on the y axis?

Charge of uniform density 12 nC/m is distributed along the x axis from x = 2.0 m to x = 5.0 m.What is the electric potential (relative to zero at infinity) at the origin (x = 0) ?

A solid conducting sphere (radius = 5.0 cm) has a charge of 0.25 nC distributed uniformly on its surface.If point A is located at the center of the sphere and point B is 15 cm from the center,what is the magnitude of the electric potential difference between these two points?

Two large parallel conducting plates are 8.0 cm apart and carry equal but opposite charges on their facing surfaces.The magnitude of the surface charge density on either of the facing surfaces is 2.0 nC/m2.Determine the magnitude of the electric potential difference between the plates.

A charge per unit length given by λ(x) = bx,where b = 12 nC/m2,is distributed along the x axis from x = +9.0 cm to x = +16 cm.If the electric potential at infinity is taken to be zero,what is the electric potential at the point P on the y axis at y = 12 cm?

A nonconducting sphere of radius 10 cm is charged uniformly with a density of 100 nC/m3.What is the magnitude of the potential difference between the center and a point 4.0 cm away?

Charge of uniform density 90 nC/m3 is distributed throughout the inside of a long nonconducting cylindrical rod (radius = 2.0 cm) .Determine the magnitude of the potential difference of point A (2.0 cm from the axis of the rod) and point B (4.0 cm from the axis) .

A charge of uniform density (0.80 nC/m) is distributed along the x axis from the origin to the point x = 10 cm.What is the electric potential (relative to zero at infinity) at a point,x = 18 cm,on the x axis?

A charge of 18 nC is uniformly distributed along the y axis from y = 3 m to y = 5 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point x = +2 m on the x axis?

A charge of 20 nC is distributed uniformly along the x axis from x = −2.0 m to x = +2.0 m.What is the electric potential (relative to zero at infinity) at the point x = 5.0 m on the x axis?

A linear charge of nonuniform density λ = bx,where b = 2.1 nC/m2,is distributed along the x axis from x = 2.0 m to x = 3.0 m.Determine the electric potential (relative to zero at infinity) of the point y = 4.0 m on the y axis.

A rod (length = 2.0 m) is uniformly charged and has a total charge of 5.0 nC.What is the electric potential (relative to zero at infinity) at a point which lies along the axis of the rod and is 3.0 m from the center of the rod?

A charge of 20 nC is distributed uniformly along the y axis from y = 0 to y = 4 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point x = +3 m on the x axis?

A charge of 10 nC is distributed uniformly along the x axis from x = −2 m to x = +3 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point x = +5 m on the x axis?

Charge of uniform linear density 6.0 nC/m is distributed along the x axis from x = 0 to x = +3 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point y = +4 m on the y axis?

Charge of uniform density (3.5 nC/m) is distributed along the circular arc shown.Determine the electric potential (relative to zero at infinity) at point P.

Charge of uniform linear density 3.0 nC/m is distributed along the x axis from x = 0 to x = 3 m.Which of the following integrals is correct for the electric potential (relative to zero at infinity) at the point x = +4 m on the x axis?

Charge of uniform density 50 nC/m³ is distributed throughout the inside of a long nonconducting cylindrical rod (radius = 5.0 cm) .Determine the magnitude of the potential difference of point A (2.0 cm from the axis of the rod) and point B (4.0 cm from the axis) .

A nonuniform linear charge distribution given by λ(x) = bx,where b is a constant,is distributed along the x axis from x = 0 to x = +L.If b = 40 nC/m² and L = 0.20 m,what is the electric potential (relative to a potential of zero at infinity) at the point y = 2L on the y axis?

Two large parallel conducting plates are 8.0 cm apart and carry equal but opposite charges on their facing surfaces.The magnitude of the surface charge density on either of the facing surfaces is 2.0 nC/m².Determine the magnitude of the electric potential difference between the plates.

A charge per unit length given by λ(x) = bx,where b = 12 nC/m²,is distributed along the x axis from x = +9.0 cm to x = +16 cm.If the electric potential at infinity is taken to be zero,what is the electric potential at the point P on the y axis at y = 12 cm?

A nonconducting sphere of radius 10 cm is charged uniformly with a density of 100 nC/m³.What is the magnitude of the potential difference between the center and a point 4.0 cm away?

Charge of uniform density 90 nC/m³ is distributed throughout the inside of a long nonconducting cylindrical rod (radius = 2.0 cm) .Determine the magnitude of the potential difference of point A (2.0 cm from the axis of the rod) and point B (4.0 cm from the axis) .

A linear charge of nonuniform density λ = bx,where b = 2.1 nC/m²,is distributed along the x axis from x = 2.0 m to x = 3.0 m.Determine the electric potential (relative to zero at infinity) of the point y = 4.0 m on the y axis.