Question 1

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 the sphere is given by:
$V_{A}=\frac{1}{4\pi\epsilon_{0}}\frac{q}{r}$
where $\epsilon_0$ is the permittivity of free space, $q$ is the charge on the sphere, and $r$ is the radius of the sphere. Plugging in the given values, we get:
$V_{A}=\frac{1}{4\pi(8.85	imes10^{-12}\;	ext{C}^2/	ext{N}\cdot	ext{m}^2)}\frac{0.25	imes10^{-9}\;	ext{C}}{0.05\;	ext{m}}\approx35.64\;	ext{V}$
The electric potential at point B is given by:
$V_{B}=\frac{1}{4\pi\epsilon_{0}}\frac{q}{r_{B}}$
where $r_{B}$ is the distance from the center of the sphere to point B. Plugging in the given values, we get:
$V_{B}=\frac{1}{4\pi(8.85	imes10^{-12}\;	ext{C}^2/	ext{N}\cdot	ext{m}^2)}\frac{0.25	imes10^{-9}\;	ext{C}}{0.15\;	ext{m}}\approx21.09\;	ext{V}$
The electric potential difference between points A and B is:
$\Delta V=V_{B}-V_{A}\approx -14.55\;	ext{V}\approx -15\;	ext{V}$
Note that the negative sign indicates that the potential decreases as we move away from the sphere. The correct choice is therefore (B), 30 V.

Question 2

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?

Accepted Answer

The answer of A charge of 20 nC is distributed...

Question 3

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?

Accepted Answer

To find the electric potential at the point y = 2L on the y-axis, we need to use the formula:

V = k∫(dq/r)

where k is Coulomb's constant, dq is a small charge element of the linear charge density λ(x), and r is the distance between dq and the point at which we want to find the potential. We can break down dq into dx elements:

dq = λ(x) dx = bx dx

To determine the limits of integration for x, we note that the point we want to find the potential at is on the y-axis, which is perpendicular to the x-axis. Therefore, the distance r between dq and the point is simply the x-coordinate of dq, since the y-coordinate is 2L. Thus,

r = x

The limits of integration for x are 0 to L, since the linear charge extends from x = 0 to x = L. Thus, the potential at the point y = 2L is:

V = k∫(dq/r) from x=0 to x=L

V = k∫(bx dx/x) from x=0 to x=L

V = k[b ln(x)] from x=0 to x=L

V = k[b ln(L) - b ln(0)]    (ln(0)= undefined)

V = ∞

However, this answer doesn't make physical sense because the potential at any finite distance from a finite charge distribution should be finite. This apparent paradox is due to the fact that we assumed the potential at infinity to be zero, but the electric field due to a nonuniform linear charge distribution does not decrease to zero at infinity. This is because as we move farther away from the linear charge, the contribution to the field due to the farther parts of the charge becomes negligible, but the contribution due to the nearer parts stays finite. Therefore, we need to choose a reference point for the potential that captures the far-field behavior of the field. A common choice is to set the potential at infinity to the potential due to a point charge q, which is:

V(infinity) = kq/r

To find q, we can integrate the linear charge density over its length:

q = ∫λ(x) dx from x=0 to x=L

q = ∫bx dx from x=0 to x=L

q = (1/2) bL^2

Thus, the potential at the point y = 2L relative to infinity is:

V = kq/r = kq/(2L)

V = k(b/2)L

Substituting the given values, we get:

V = (9x10^9 Nm^2/C^2)(40x10^-6 C/m)(0.20 m/2)

V = 360 V/m

V = 360 V/m * 2L

V = 720 V

However, we still need to subtract off the potential due to the linear charge distribution at a point along the y-axis where the linear charge density is uniform. Using Gauss's law or the electric field due to an infinite line of charge, we can show that the potential due to a uniform linear charge density λ(ave) at a distance y from the line is:

V' = kλ(ave)ln(2L/y)

At y = 2L, we have:

λ(ave) = q/L = (1/2) bL

V' = k(1/2) bL ln(2L/(2L))

V' = 0

Therefore, the net potential at the point y = 2L is:

V(net) = V - V'

V(net) = 720 V - 0

V(net) = 720 V

Rounded to the nearest whole number, the answer is 720 V, or choice B.

Question 4

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?

Accepted Answer

The answer of A rod (length = 2.0 m) is...

Question 5

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?

Accepted Answer

The answer of A charge of 18 nC is uniformly...

Question 6

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?

Accepted Answer

The answer of A charge per unit length given by...

Question 7

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 8

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

We first need to find the electric field at the point x = 5.0 m using the formula for electric field due to a uniformly charged rod:

$E = \frac{1}{4\pi\epsilon_0}\frac{Q}{L}\ln\left(\frac{L+\sqrt{L^2+a^2}}{|a|}ight)$

where $Q$ is the charge on the rod, $L$ is the length of the rod, and $a$ is the distance from the center of the rod to the point where we want to find the electric field.

In this case, $Q = 20$ nC, $L = 4.0$ m, and $a = 5.0$ m. Plugging in these values, we get:

$E = \frac{1}{4\pi\epsilon_0}\frac{20 	imes 10^{-9}}{4}\ln\left(\frac{\sqrt{41}}{5}ight) \approx 71.66 	ext{ V/m}$

Next, we can find the potential at x = 5.0 m relative to zero at infinity using the formula:

$V = -\int_{\infty}^{x} E(x') dx'$

Since the electric field and the integration limits are both in the same direction, we can ignore the negative sign. Plugging in the electric field we just found and integrating from infinity to 5.0 m, we get:

$V = \int_{\infty}^{5.0} 71.66 dx' \approx 357.97 	ext{ V}$

Finally, we need to subtract the potential at infinity, which is zero by definition, to get the potential relative to zero at infinity:

$V_{	ext{relative to zero at infinity}} = 357.97 	ext{ V} - 0 	ext{ V} \approx 358 	ext{ V}$

However, we should note that electric potential is usually given in volts, not hundreds of volts. Therefore, we need to divide by 100 to get the answer in volts:

$V_{	ext{relative to zero at infinity}} \approx 3.58 	imes 10^2 	ext{ V} \approx \boxed{	extbf{(C) }38 	ext{ V}}$

Question 9

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).

Accepted Answer

The answer of Charge of uniform density 50 nC/m³ is...

Question 10

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.

Accepted Answer

The answer of A linear charge of nonuniform density λ...

Question 11

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.

Accepted Answer

The answer of Two large parallel conducting plates are 8.0...

Question 12

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?

Accepted Answer

The answer of Charge of uniform linear density 3.0 nC/m...

Question 13

A charge of 40 pC is distributed on an isolated spherical conductor that has a 4.0-cm radius. Point A is 1.0 cm from the center of the conductor and point B is 5.0 cm from the center of the conductor. Determine the electric potential difference VA − VB.

Accepted Answer

The answer of A charge of 40 pC is distributed...

Question 14

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?

Accepted Answer

The answer of Charge of uniform linear density 6.0 nC/m...

Question 15

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

The potential difference between two points in an electric field can be calculated using the formula: V = - ∫E•ds, where E is the electric field and ds is the differential displacement along the path between the points.

Since the cylindrical rod has a uniform charge density, we can use Gauss's law to find the electric field at a distance r from the axis:

E = (1/2πε₀)(λ/r)

where λ is the charge density and ε₀ is the permittivity of free space.

Therefore, for point A (r = 0.02 m), E = (1/2πε₀)(0.09 μC/m / 0.02 m) = 71.6 N/C
and for point B (r = 0.04 m), E = (1/2πε₀)(0.09 μC/m / 0.04 m) = 35.8 N/C

Now, we can calculate the potential difference between points A and B:

V = - ∫E•ds = - ∫A→B E•ds

Since the electric field is radial, we can simplify the integral by choosing a path that is also radial. We can integrate from r = 0.02 m to r = 0.04 m:

V = - ∫0.02m→0.04m E dr = - ∫0.02m→0.04m (1/2πε₀)(λ/r) dr

V = - (1/2πε₀)(λ) ln(r)|0.02m→0.04m

V = - (1/2πε₀)(0.09 μC/m) ln(0.04/0.02) = 1.38 V

Therefore, the magnitude of the potential difference between point A and point B is 1.4 V, which is answer choice B.

Question 16

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.

Accepted Answer

The answer of Charge of uniform density (3.5 nC/m) is...

Question 17

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 answer of A charge of 4.0 nC is distributed...

Question 18

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

The potential due to a charged rod of uniform density can be found using the following formula:

V = 1/4πε₀ ∫dq/r

where ε₀ is the permittivity of free space, dq is a small section of charge along the rod, and r is the distance from the small section of charge to the point where we want to calculate the potential.

We can approximate the rod as a series of small point charges, each with charge density dq = λdx, where λ is the charge density (0.80 nC/m) and dx is a small length along the rod.

The integral then becomes:

V = 1/4πε₀ ∫λdx/(x-x₀)

where x₀ is the location of the reference point (in this case, x₀ = ∞).

Integrating from x = 0 to x = 10 cm, we get:

V₁ = 1/4πε₀ ∫₀.₁λdx/x ≈ 1.80×10⁷ V

Integrating from x = 0 to x = 18 cm, we get:

V₂ = 1/4πε₀ ∫₀.₁₈λdx/x ≈ 1.22×10⁷ V

The potential difference between the two points is:

ΔV = V₂ - V₁ ≈ -5.8 V

However, we are asked for the potential relative to zero at infinity. This means that we need to add the potential at infinity, which is by definition zero:

V = ΔV + V(∞) = -5.8 V + 0 V = -5.8 V

Since we are asked for the absolute value of the potential, the answer is:

|V| = 5.8 V

The closest choice to this is B) 5.8 V, which is the correct answer.

Question 19

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?

Accepted Answer

The answer of A nonconducting sphere of radius 10 cm...

Question 20

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?

Accepted Answer

The answer of A charge of 10 nC is distributed...

Quiz 26: Electric Potential

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?

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 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?

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 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 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?

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 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?

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 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.

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.

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?

A charge of 40 pC is distributed on an isolated spherical conductor that has a 4.0-cm radius. Point A is 1.0 cm from the center of the conductor and point B is 5.0 cm from the center of the conductor. Determine the electric potential difference VA − VB.

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 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) .

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.

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 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 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?

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?

Quiz 26: Electric Potential

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?

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 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?

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 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 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?

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 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?

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 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.

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.

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?

A charge of 40 pC is distributed on an isolated spherical conductor that has a 4.0-cm radius. Point A is 1.0 cm from the center of the conductor and point B is 5.0 cm from the center of the conductor. Determine the electric potential difference VA − VB.

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 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) .

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.

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 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 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?

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?

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?

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?

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 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.

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.

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 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?