Question

Calculate the electric dipole moment of a system comprising a charge +q distributed uniformly on a semicircular arc of radius R and a point charge -q kept at its center.

• A. $\frac{2Rq}{\pi}$
• B. $\frac{Rq}{\pi}$
• C. $\frac{4Rq}{\pi}$
• D. 0

Answer 1

Answer: (A)

$\frac{2Rq}{\pi}$

Answer 2

The electric dipole moment of a system is given by $\vec{p} = \sum q_i \vec{r_i}$ for discrete charges or $\vec{p} = \int \vec{r} dq$ for continuous charge distributions. The point charge $-q$ at the origin contributes zero to the dipole moment. For the semicircular arc, consider a differential charge element $dq = \frac{q}{\pi}d\theta$ at position $(R\cos\theta, R\sin\theta)$. Integrate $\vec{r}dq$ from $\theta=0$ to $\theta=\pi$. The x-component integral $\int_0^\pi R\cos\theta \frac{q}{\pi}d\theta$ evaluates to zero due to symmetry. The y-component integral $\int_0^\pi R\sin\theta \frac{q}{\pi}d\theta$ evaluates to $\frac{Rq}{\pi} [-\cos\theta]_0^\pi = \frac{Rq}{\pi} (1 - (-1)) = \frac{2Rq}{\pi}$. Thus, the total dipole moment is $\frac{2Rq}{\pi}$ in the y-direction, and its magnitude is $\frac{2Rq}{\pi}$.