A circle of radius a has a linear charge dens | vedclass

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

Question

(A) As shown in the figure,let us consider an element of angular width $d 	heta$ at an angle $	heta$.The length of this element is $dl = a d 	heta$

Vedclass · Accepted Answer

$\pi a \lambda_0$

Vedclass · Answer

(A) As shown in the figure,let us consider an element of angular width $d 	heta$ at an angle $	heta$.The length of this element is $dl = a d 	heta$.The charge on this element is $dq = \lambda dl = (\lambda_0 \cos^2 	heta) (a d 	heta)$.The total charge $Q$ on the circle is obtained by integrating $dq$ over the entire circumference from $0$ to $2 \pi$:$Q = \int_{0}^{2 \pi} \lambda_0 \cos^2 	heta a d 	heta$$Q = a \lambda_0 \int_{0}^{2 \pi} \cos^2 	heta d 	heta$Using the identity $\cos^2 	heta = \frac{1 + \cos 2 	heta}{2}$:$Q = a \lambda_0 \int_{0}^{2 \pi} \frac{1 + \cos 2 	heta}{2} d 	heta$$Q = \frac{a \lambda_0}{2} [	heta + \frac{\sin 2 	heta}{2}]_{0}^{2 \pi}$$Q = \frac{a \lambda_0}{2} [(2 \pi + 0) - (0 + 0)]$$Q = \frac{a \lambda_0}{2} (2 \pi) = \pi a \lambda_0$Thus,the total charge is $\pi a \lambda_0$.

$A$ circle of radius $a$ has a linear charge density given by $\lambda = \lambda_0 \cos^2 \theta$ on its circumference. What will be the total charge on the circle?

Similar Questions

The surface charge density $\sigma$ on a charged conducting sphere of radius $R$ in terms of the electric field intensity $E$ at a distance $r$ $(r > R)$ in free space is (where $\varepsilon_{0}$ is the permittivity of free space):

Charges $Q, 2Q$ and $-Q$ are given to three concentric conducting shells $A, B$ and $C$ respectively as shown. The ratio of charges on the inner and outer surfaces of shell $C$ will be:

The figure shows a charged conductor resting on an insulating stand. If at point $P$ the charge density is $\sigma$,the potential is $V$,and the electric field strength is $E$,what are the values of these quantities at point $Q$?

If a metal cube of side $5 \, cm$ has a charge of $6 \, \mu C$,then the surface charge density is

Three concentric metallic spherical shells of radii $R, 2R, 3R$ are given charges $Q_1, Q_2, Q_3$ respectively. It is found that the surface charge densities on the outer surfaces of the shells are equal. Then,the ratio of the charges given to the shells,$Q_1 : Q_2 : Q_3$,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module