A charge +q is distributed over a thin ring o | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The total charge $q_{	ext{total}}$ on the ring is given by the integral of the charge density $\lambda$ over the circumference:$q_{	ext{total}}

Vedclass · Accepted Answer

equal to $qQ / 4\pi \varepsilon_0 r$

Vedclass · Answer

(B) The total charge $q_{	ext{total}}$ on the ring is given by the integral of the charge density $\lambda$ over the circumference:$q_{	ext{total}} = \int_0^{2\pi} \lambda (r d	heta) = \int_0^{2\pi} \frac{q \sin^2 	heta}{\pi r} (r d	heta) = \frac{q}{\pi} \int_0^{2\pi} \sin^2 	heta d	heta$Using the identity $\sin^2 	heta = (1 - \cos 2	heta) / 2$:$q_{	ext{total}} = \frac{q}{\pi} \int_0^{2\pi} \frac{1 - \cos 2	heta}{2} d	heta = \frac{q}{2\pi} [	heta - \frac{\sin 2	heta}{2}]_0^{2\pi} = \frac{q}{2\pi} (2\pi) = q$Since all points on the ring are at a distance $r$ from the centre,the electric potential $V$ at the centre is:$V = \frac{1}{4\pi \varepsilon_0} \int \frac{dq}{r} = \frac{1}{4\pi \varepsilon_0 r} \int dq = \frac{q_{	ext{total}}}{4\pi \varepsilon_0 r} = \frac{q}{4\pi \varepsilon_0 r}$The work done by the electric force in moving a charge $Q$ from the centre to infinity is $W = Q(V_{	ext{centre}} - V_{\infty})$. Since $V_{\infty} = 0$:$W = Q \cdot V = \frac{qQ}{4\pi \varepsilon_0 r}$

Similar Questions

An electron (charge = $1.6 \times 10^{-19} \text{ C}$) is accelerated through a potential of $100,000 \text{ V}$. The energy acquired by the electron is:

Two rings of radius $R$ are placed coaxially at a distance $R$ apart. They carry charges $Q_1$ and $Q_2$ respectively. How much work is required to move a charge $q$ from the center of one ring to the center of the other?

$A$ charge of total amount $Q$ is distributed over two concentric hollow spheres of radii $r$ and $R$ $(R > r)$ such that the surface charge densities on the two spheres are equal. The electric potential at the common centre is

$A$ regular hexagon of side $5 \, cm$ has a charge $10 \, \mu C$ at each of its vertices. The potential at the centre of the hexagon is

$A$ total charge $Q$ is distributed between two concentric spherical shells of radii $r$ and $R$ $(R > r)$ such that their surface charge densities are equal. What is the electric potential at their common center?

Vedclass Test Series

Exam Paper Generator

Online Exam Module