# Test 4

**Test 4A:** Questions 1–3.  
**Test 4B:** Questions 4–6.  
**Max point:** 100 + 100.

**NB!** Small sketches may improve the answers in questions 1, 3, 4, 5, 6.

## 1. (30p)

What is the work done by the electric force, when $q_0$ is moved from $a$ to $b$?

## 2. (15p)

Please write the condition for conservation of the total energy.

## 3. (55p)

When $q_0$ moves from $a$ to $b$ in an E-field from a point charge $q$ (“Central field”).

**a. (30p)** Please derive the work done by the electric force.

**b. (25p)** The work in above is equal to the drop in the electric potential energy
$U_{\mathrm{e\,pot}\,a} - U_{\mathrm{e\,pot}\,b}$.
What is $U_{\mathrm{e\,pot}\,a}$ and $U_{\mathrm{e\,pot}\,b}$ if $b \to \infty$?

## 4. (25p)

Bringing charges together.

**a. (15p)** What is the total electric potential energy if $q_1, q_2, q_3$ are brought together from infinite?

**b. (10p)** What is the additional electric potential energy if $q_0$ is brought together with $q_1, q_2, q_3$?

## 5. (25p)

When $q_0$ moves from $a$ to $b$ in a uniform E-field.

**a. (15p)** Please derive the work done by the electric force.

**b. (10p)** What is $U_{\mathrm{pot}\,a}$ and $U_{\mathrm{pot}\,b}$?

## 6. (50p)

The electric dipole in a uniform E-field.

**a. (10p)** Please derive the torque vector.

Prove, that the magnitude is

$$
\tau = p_e E \sin(\theta),
$$

where $\theta = \angle(\vec{p}_e, \vec{E})$.

**b. (30p)** Please derive the work $W_{ab}$ done by the electric force when the electric dipole rotates from $\theta_a$ to $\theta_b$ and what is $U_{\mathrm{e\,pot}\,a}$ and $U_{\mathrm{e\,pot}\,b}$?

You don’t have to derive $dW = -\tau\,d\theta$.

**c. (10p)** Explain why $\theta_a = 90^\circ$ can be used as a reference position for the electric potential energy and write the result for $U_{\mathrm{e\,pot}\,a}$.

Now find the general expression of $U_{\mathrm{e\,pot}}$ for the electric dipole, also written as a scalar product.