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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.