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# 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.
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