Question 1

A point charge $$q is $$placed at the origin. What is the expression for the electric field $$\vec{E}(\vec{r}) at a $$point $$\vec{r} in $$space?

Accepted Answer

$$\vec{E}(\vec{r}) = k_e \frac{q}{r^2$} \frac{\vec{r}}{r}$$

Question 2

A point charge $$q is $$located at position $$\vec{r}_0. $$What is the electric field at point $$\vec{r}$$?

Accepted Answer

$$\vec{E}(\vec{r}) = k_e \frac{q}{|\vec{r} - \vec{r}_0|^2} \frac{\vec{r} - \vec{r}_0}{|\vec{r} - \vec{r}_0|}$$

Question 3

For a continuous charge distribution, which integral gives the electric field at point $$\vec{r}$$?

Accepted Answer

$$\vec{E}(\vec{r}) = k_e \int \frac{dq(\vec{r}')}{|\vec{r} - \vec{r}'|^3} (\vec{r} - \vec{r}')$$

Question 4

A ring of radius $$r$$ carries a total charge $$q. $$What is the electric field $$\vec{E}(x) at a $$point $$x on $$its axis of symmetry?

Accepted Answer

$$\vec{E}(x) = k_e \frac{q x}{(x^2$ + r^2$)^{3/2}$$}$

Question 5

A thin disk with a circular hole in the center has uniform surface charge density $$\sigma. $$The inner radius is $$R_1$$ and the outer radius is $$R_2. $$What is the electric field $$\vec{E}(x) at a $$point $$x on $$its axis of symmetry?

Accepted Answer

$$\vec{E}(x) = 2 \pi k_e \sigma \left[ \left(1 - \frac{x}{\sqrt{x^2$ + R_2$^2}} \right) - \left(1 - \frac{x}{\sqrt{x^2$ + R_1$^2}} \right) \right]$$

Question 6

A rod of length $$L$$ has a positive charge $$Q$$ uniformly distributed along its length. What is the linear charge density $$\lambda$$?

Accepted Answer

$$\lambda = \frac{Q}{L}$$

Question 7

For a continuous charge distribution, which expression represents the volume charge density?

Accepted Answer

$$\rho(\vec{r}) = \frac{dq(\vec{r})}{dV(\vec{r})}$$

Question 8

A dipole $$\vec{p}_e = q \vec{d} is $$centered at the origin and oriented along the y-axis. For $$|\vec{r}| \gg d$$, what is the approximate electric field at point $$\vec{r}$$?

Accepted Answer

$$\vec{E}(\vec{r}) \approx k_e \frac{q d}{|\vec{r}|^3}$$

Question 9

Which of the following is the correct expression for the surface charge density $$\sigma(\vec{r})$$?

Accepted Answer

$$\sigma(\vec{r}) = \frac{dq(\vec{r})}{dA(\vec{r})}$$

Question 10

Which integration trick would be most useful for evaluating the electric field of a ring of charge along its axis?

Accepted Answer

Using symmetry to reduce the vector components and integrating over the ring's circumference

Question 11

A rod of length $$L$$ with total charge $$Q is $$placed along the x-axis. What is the electric field at a point $$y \u003e L/2 on $$the y-axis?

Accepted Answer

$$\vec{E}(y) = k_e \frac{Q}{L} \int_{-L/2}^{L/2} \frac{y}{(x^2$ + y^2$)^{3/2}$$} dx$

Question 12

If a rod of length $$L$$ with uniform charge $$Q$$ becomes infinitely long, what is the electric field at a point $$x on $$the x-axis?

Accepted Answer

$$\vec{E}(x) = 2 k_e \lambda \frac{1}{x}$$

Question 13

Which of the following is NOT a common type of charge distribution?

Accepted Answer

Mass charge density $$\mu(\vec{r})$$

Question 14

Which indefinite integral is useful for calculating the electric field from a ring of charge along its axis?

Accepted Answer

$$\int dx \frac{x}{(x^2$ + y^2$)^{3/2}$$} = -\frac{1}{\sqrt{$$x^2$$ + $$y^2$$}}$

Question 15

Which physical scenario would require the use of a continuous charge distribution rather than point charges?

Accepted Answer

A charged metal rod whose length is much greater than the spacing between atoms

Question 16

If the electric field at a point is $$\vec{E}(\vec{r})$$, what is the force on a charge $$q$$ placed at that point?

Accepted Answer

$$\vec{F} = q \vec{E}(\vec{r})$$

Question 17

Which method is most effective for simplifying the calculation of the electric field from a symmetric charge distribution?

Accepted Answer

Exploiting symmetry to reduce the number of variables in the integral

Question 18

A surface has a uniform charge density $$\sigma. $$What is the total charge $$Q on a $$surface area $$A$$?

Accepted Answer

$$Q = \sigma A$$

Electric Field Calculation: Point Charges and Continuous Distributions

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