(a) (i) Equipotential Surfaces for an Electric Dipole
The equipotential surfaces for an electric dipole are surfaces where the electric potential is constant. These surfaces are crowded near the charges and become more spread out as the distance from the dipole increases. They are symmetric about the axis of the dipole.
[Note: A diagram showing closed loops around each charge, with the loops becoming larger and more distorted as they encompass both charges, is required for full marks.]
(a) (ii) Potential Energy of a System of Two Charges in an External Field
Step 1 (Governing Law): [1 Mark]
The potential energy $U$ of a system of two charges $q_1$ and $q_2$ in an external field $\vec{E}$ is the sum of the work done in bringing the charges from infinity to their respective positions $\vec{r}_1$ and $\vec{r}_2$ and the work done in bringing them against their mutual electrostatic force.
Step 2 (Mathematical Derivation): [1 Mark]
- Work done to bring $q_1$ to $\vec{r}_1$: $W_1 = q_1 V(\vec{r}_1)$
- Work done to bring $q_2$ to $\vec{r}_2$: $W_2 = q_2 V(\vec{r}_2) + \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{|\vec{r}_1 - \vec{r}_2|}$
Step 3 (Final Expression): [1 Mark]
Total Potential Energy $U = W_1 + W_2$:
$$U = q_1 V(\vec{r}_1) + q_2 V(\vec{r}_2) + \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{|\vec{r}_1 - \vec{r}_2|}$$(a) (iii) Numerical Calculation
Step 1 (Formula & Given Data): [1 Mark]
The potential energy of a dipole in an electric field is given by $U = -\vec{p} \cdot \vec{E} = -pE \cos\theta$.
- $p = 10^{-30} \text{ Cm}$
- $E = 10^5 \text{ V/m}$
- Initial angle $\theta_1 = 0^\circ$
- Final angle $\theta_2 = 60^\circ$
Step 2 (Mathematical Working): [0.5 Mark]
Change in potential energy $\Delta U = U_f - U_i$:
$$\Delta U = (-pE \cos 60^\circ) - (-pE \cos 0^\circ)$$ $$\Delta U = pE (\cos 0^\circ - \cos 60^\circ)$$ $$\Delta U = (10^{-30})(10^5) \left(1 - \frac{1}{2}\right)$$ $$\Delta U = 10^{-25} \times 0.5$$Step 3 (Final Answer): [0.5 Mark]
$$\Delta U = 5 \times 10^{-26} \text{ J}$$💡 EL10 Concept Insight: Think of the dipole like a compass needle in a magnetic field; it wants to align