# Charge Distributions#

Two thin rings of radius $$R$$ and charges $$Q_1 = Q$$ and $$Q_2 = -Q$$ are centred along the $$x$$-axis as shown in the figure below.

## Part 1#

Write an expression in terms of $$x$$, $$d$$ and $$R$$ for the distance, $$r_1$$, for the distance of any point on ring 1 from point $$P$$. Note that this is a right-angled triangle so you can use the pythagorean theorem.

Use the following table as a reference for each variable:

For

Use

$$R$$

R

$$d$$

d

$$x$$

x

$$k$$

k

$$Q$$

Q

If the answer is 0, enter “zero”.

## Part 2#

Write an expression for the electric potential at point $$P$$ due to the two thin rings in terms of $$Q$$, $$R$$, $$d$$ and $$x$$.

Use the following table as a reference for each variable:

For

Use

$$R$$

R

$$d$$

d

$$x$$

x

$$k$$

k

$$Q$$

Q

If the answer is 0, enter “zero”.

## Part 3#

From your expression for the potential energy find the electric field at point $$P$$. Note that it points along the $$x$$-axis, and remember to use the chain rule.

Use the following table as a reference for each variable:

For

Use

$$R$$

R

$$d$$

d

$$x$$

x

$$k$$

k

$$Q$$

Q

If the answer is 0, enter “zero”.

## Part 4#

When $$x = -\frac{d}{2}$$ evaluate the electric potential.

Use the following table as a reference for each variable:

For

Use

$$R$$

R

$$d$$

d

$$x$$

x

$$k$$

k

$$Q$$

Q

If the answer is 0, enter “zero”.

## Part 5#

When $$x = -\frac{d}{2}$$ evaluate the electric field.

Use the following table as a reference for each variable:

For

Use

$$R$$

R

$$d$$

d

$$x$$

x

$$k$$

k

$$Q$$

Q

If the answer is 0, enter “zero”.