Question 1

Eight point charges, each of $$100\ \mathrm{nC}$$, are placed at the corners of a cube with side length $$10\ \mathrm{cm}. If $$the electric potential is defined to be zero at infinity, what is the electric potential at the center of the cube? Use $$\$$varepsilon_0$$ = 8.854 	imes 10$$^{-12}$$\ \mathrm{C^2$/(N\cdot m^2$)}$$.

$$V = \sum_i \frac{q_i}{4\pi\$$varepsilon_0$$ r_i}$$, where $$r_i is $$the distance from each charge to the center.

Accepted Answer

$$8.3 	imes 10^4$ Volts

Question 2

If the side length of the cube in the previous problem is doubled, but the charge on each corner remains $$100\ \mathrm{nC}$$, what happens to the electric potential at the center of the cube?

Accepted Answer

It is reduced by a factor of $$1/\sqrt{2}.$$

Question 3

Which of the following best explains why the potential at the center of the cube is the sum of the potentials from each charge, rather than the vector sum?

Accepted Answer

Potential is a scalar quantity, so it adds algebraically.

Question 4

Given the formula $$V = \sum_i \frac{q_i}{4\pi\$$varepsilon_0$$ r_i}$$, what is the value of $$r_i$$ for each charge in the cube problem?

Accepted Answer

$$\frac{\sqrt{3}\ \mathrm{m}}{20}$$

Question 5

If one of the eight charges is removed from the cube, what happens to the potential at the center?

Accepted Answer

It decreases by $$1/8 of $$the original value.

Question 6

Suppose the charges at the corners of the cube are alternately positive and negative (i.e., four positive and four negative, arranged symmetrically). What is the electric potential at the center of the cube?

Accepted Answer

Zero, due to symmetry and cancellation.

Question 7

If the charges are doubled to $$200\ \mathrm{nC}$$ each, what happens to the potential at the center of the cube?

Accepted Answer

It doubles.

Question 8

Which physical constant is necessary to calculate the electric potential in this scenario?

Accepted Answer

The permittivity of free space, $$\$$varepsilon_0$$

Question 9

If the cube is placed in a vacuum instead of air, how does the potential at the center change?

Accepted Answer

It remains the same, since air and vacuum have nearly identical $$\$$varepsilon_0$$.

Question 10

A student claims that the electric field at the center of the cube is nonzero. Is this correct?

Accepted Answer

No, the electric field is zero due to symmetry.

Electric Potential at the Center of a Cube with Point Charges

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