# Continuous Charge Distributions#

A thin ring of radius $$R$$ lies in the $$xy$$ plane, centred on the $$z$$-axis. The ring has four uncharged insulating pieces. (dark patches of figure below) and four metallic ars. Two of these metallic arcs have charge $$q_1$$ and two have charge $$q_2$$, symmetrically placed about ($$x$$, $$y$$, $$z$$) = (0,0,0) as shown in the figure below.

## Part 1#

Write the small element of the electric potential $$dV_1$$ at heigh $$z$$ along the $$z$$-axis due to a small element of charge $$dq_1$$ as shown in the figure above. Hint: assume the small element of charge is small enough to treat as a point charge.

Use the following table as a reference for each variable:

For

Use

$$R$$

R

$$K$$

K

$$dq_1$$

dq1

$$dq_2$$

dq2

$$q_1$$

q1

$$q_2$$

q2

$$z$$

z

If the answer is 0, enter “zero”.

## Part 2#

Write the small element of the electric potential $$dV_2$$ at heigh $$z$$ along the $$z$$-axis due to a small element of charge $$dq_2$$ as shown in the figure above.

Use the following table as a reference for each variable:

For

Use

$$R$$

R

$$K$$

K

$$dq_1$$

dq1

$$dq_2$$

dq2

$$q_1$$

q1

$$q_2$$

q2

$$z$$

z

If the answer is 0, enter “zero”.

## Part 3#

For $$any$$ small element of charge $$dq_1$$ on either patch of charge $$q_1$$, would you get the result found in (a)?

• No you would get the same result, as the charge $$dq_1$$ would be at a different distance from the point of interest at ($$x$$, $$y$$, $$z$$) = (0, 0, $$z$$)

• Yes you would get the same result, as the charge $$dq_1$$ would be at the same distance from the point of interest at ($$x$$, $$y$$, $$z$$) = (0, 0, $$z$$)

• Yes you would get the same result, as the charge $$dq_1$$ would be at the same distance from the point of interest at ($$x$$, $$y$$, $$z$$) = ($$x$$, 0, 0)

• No you would get the same result, as the charge $$dq_1$$ would be at a different distance from the point of interest at ($$x$$, $$y$$, $$z$$) = ($$x$$, 0, 0)

## Part 4#

Integrate your expressions from parts (a) and (b) to find the total electric potential due to this ring along the $$z$$-axis at $$z$$ = 3m.

Use the following table as a reference for each variable:

For

Use

$$R$$

R

$$K$$

K

$$dq_1$$

dq1

$$dq_2$$

dq2

$$q_1$$

q1

$$q_2$$

q2

$$z$$

z

If the answer is 0, enter “zero”.

## Part 5#

Assuming that the charged patches are of equal sizes and on opposite sides of the ring, as shown in Fig. 1, explain what direction the electric field of this ring would have to point in and why.

• The equal charges are uniformly distributed on the opposite sides of the x-axis, so any components parallel to the x-axis will add to zero. The electric field must be perpendicular to the x-axis

• The equal charges are uniformly distributed on the opposite sides of the z-axis, so any components parallel to the z-axis will add to zero. The electric field must be perpendicular to the z-axis

• The equal charges are uniformly distributed on the opposite sides of the x-axis, so any components perpendicular to the x-axis will add to zero. The electric field must be parallel (or antiparallel) to the x-axis

• The equal charges are uniformly distributed on the opposite sides of the z-axis, so any components perpendicular to the z-axis will add to zero. The electric field must be parallel (or antiparallel) to the z-axis

## Part 6#

Find the $$z$$-component of the electric field due to this ring at height $$z$$.

Use the following table as a reference for each variable:

For

Use

$$R$$

R

$$K$$

K

$$dq_1$$

dq1

$$dq_2$$

dq2

$$q_1$$

q1

$$q_2$$

q2

$$z$$

z

If the answer is 0, enter “zero”.

## Part 7#

If $$q_1$$ = -2$$\mu C$$ and $$q_2$$ = 4$$\mu C$$, $$R$$ = 4m and $$z$$ = 6m, find $$V$$.

Note: $$\mu = 10^{-6}$$ and $$K$$ = 9.0 $$\times 10^9 \frac{\text{N m}^2}{\text{C}^2}$$

If $$q_1$$ = -2$$\mu C$$ and $$q_2$$ = 4$$\mu C$$, $$R$$ = 4m and $$z$$ = 6m, find $$E_z$$.
Note: $$\mu = 10^{-6}$$ and $$K$$ = 9.0 $$\times 10^9 \frac{\text{N m}^2}{\text{C}^2}$$