Question 1

A charge $$q_1 = +5$$\ \mathrm{nC}$$ is at the origin, q_2$ = -3\ \mathrm{nC} is at$$ $$x_2 = 4$$\ \mathrm{cm}$$, and q_3$ = -2\ \mathrm{nC} is at$$ $$x_3 = 3$$\ \mathrm{cm}$$. What is the direction of the net electric force on q_3$ due to q_1$ and q_2$?

Accepted Answer

To the left along the negative $$x-$$axis

Question 2

Given the same arrangement as above, calculate the magnitude of the net electric force on $$q_3 at$$ $$x_3 = 3$$\ \mathrm{cm}$$ using Coulomb's law. Use k = 9$$ 	imes $$10^9$$\ \mathrm{N\cdot $$m^2/C^2$}$.

Accepted Answer

$$640\ \mu\mathrm{N}$$

Question 3

For the three-charge system, at what location along the $$x-$$axis does the net electric force on $$q_3$$ become zero?

Accepted Answer

$$x_3 = 13.309$$\ \mathrm{cm}$$

Question 4

A rod of length $$2a$$ has total charge $$Q$$ uniformly distributed along its length and is centered at the origin on the $$x-$$axis. What is the expression for the line charge density $$\lambda$$?

Accepted Answer

$$\lambda = \dfrac{Q}{2a}$$

Question 5

For the same rod, what is the infinitesimal charge $$dQ on a $$segment of length $$dx$$?

Accepted Answer

$$dQ = \lambda\, dx = \dfrac{Q}{2a} dx$$

Question 6

What is the expression for the infinitesimal electric field $$d\vec{E} at a $$point $$x_0 on $$the $$x-$$axis due to a segment $$dx of $$the rod located at position $$x$$?

Accepted Answer

$$d\vec{E} = k\lambda \dfrac{dx}{(x_0$ - x)^2$} \hat{i}$$

Question 7

Integrate the infinitesimal electric field to find the net electric field $$\vec{E}(x_0$) at a $$point $$x_0 to $$the right of the rod. Which of the following is the correct expression?

Accepted Answer

$$\vec{E}(x_0$) = kQ \dfrac{1}{x_0$^2 - a^2$} \hat{i}$$

Question 8

If $$x_0$$ \gg a$$, what does the electric field $$\vec{E}($$x_0$$)$$ due to the rod reduce to?

Accepted Answer

$$\vec{E}(x_0$) = kQ \dfrac{1}{x_0$^2} \hat{i}$$

Question 9

Why does the electric field of the rod at large distances resemble that of a point charge?

Accepted Answer

Because at large distances, the rod's length becomes negligible compared to the distance, so its field approximates that of a point charge.

Question 10

A charge $$q_3 is $$moved to $$(3\ \mathrm{cm}, 2\ \mathrm{cm}). $$What is the unit vector $$\hat{r}_{13}$$ from $$q_1 at $$the origin to $$q_3$$?

Accepted Answer

$$\hat{r}_{13} = 0.832\hat{i} + 0.555\hat{j}$$

Question 11

Given $$q_1 = 5$$\ \mathrm{nC}$$ at (0$$,0)$$ and q_3$ = -2\ \mathrm{nC} at$$ $$(3,2)\ \mathrm{cm}$$, what is the magnitude of the electric field $$\vec{E}_{13} at$$ $$q_3$$ due to $$q_1$$?

Accepted Answer

$$3.462\ \mathrm{N/C}$$

Question 12

Suppose you want to check the correctness of your electric field formula for the rod by considering the limit $$x_0$$ \gg a$$. What is the main reason for performing this check?

Accepted Answer

To ensure the formula reduces to the field of a point charge at large distances, confirming its validity.

Question 13

If the rod has total charge $$Q = 10\ \mu\mathrm{C}$$ and length $$2a = 20\ \mathrm{cm}$$, what is the line charge density $$\lambda$$?

Accepted Answer

$$\lambda = 50\ \mu\mathrm{C/m}$$

Question 14

A student claims that the net electric force on $$q_3$$ can be zero if it is placed between $$q_1$$ and $$q_2. Is $$this correct? Why or why not?

Accepted Answer

No, because the net force will always point left and never be zero in that region.

Question 15

For the rod, what is the correct substitution to simplify the integral for the electric field at $$x_0$$?

Accepted Answer

$$u = x_0$ - x$$, $$du = -dx$$

Question 16

If $$q_3 is at$$ $$(3,2)\ \mathrm{cm}$$, what is the net electric field $$\vec{E}_3 at $$that point due to $$q_1$$ and $$q_2$$?

Accepted Answer

$$\vec{E}_3 = 4.490\hat{i} - 1.300\hat{j}\ \mathrm{N/C}$$

Question 17

A rod of length $$2a$$ and charge $$Q is $$centered at the origin. If you double the length of the rod but keep $$Q$$ constant, what happens to the line charge density $$\lambda$$?

Accepted Answer

$$\lambda is $$halved

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