Surface Integral

Q. Charge $Q$ is placed inside cone as shown in figure such that $\theta ={60}^{\circ }$. Flux of electric field associated with curved surface of cone is.

1. $\frac{Q}{4{\epsilon }_0}$
2. $\frac{Q}{2{\epsilon }_0}$
3. $\frac{3Q}{4{\epsilon }_0}$
4. $\frac{Q}{3{\epsilon }_0}$

Q. A point charge $q_1$ is placed inside cavity $1$ and another point charge $q_2$ is inside cavity $2$. If a point charge $q$ is placed outside the conductor then the charge on outer surface of the conductor would be

1. $Q+q_1+q_2$ and would be distributed uniformly.
2. $Q+q_1+q_2$ and it is non-uniformly distributed.
3. $Q+q_1+q_2$ and the distribution depends upon the locations of $q_1,q_2$ and $q$.
4. $Q+q_1+q_2$ and its uniform or non-uniform distribution depending upon locations of $q_1$ and $q_2$.

Q. A charge $Q$ is placed symetrically at the centre of a closed cylinder as shown in the figure. The flux associated with the curved surface of the cylinder is

1. $\frac{2Q}{{\epsilon }_0}\cos \alpha$

2. $\frac{Q}{2{\epsilon }_0}\cos \alpha$

3. $\frac{Q}{{\epsilon }_0}\cos \alpha$

4. $\frac{Q}{{\epsilon }_0}\sin \alpha$

Q. A long string with a charge of $\lambda$ per unit length passes through an imaginary cube of edge $l$. The maximum possible flux of the electric field through the cube will be

1. $\frac{\lambda l}{{ϵ}_0}$
2. $\frac{\sqrt{2}\lambda l}{{ϵ}_0}$
3. $\frac{6\lambda l^2}{{ϵ}_0}$
4. $\frac{\sqrt{3}\lambda l}{{ϵ}_0}$

Q. A positive charge $q$ is placed in front of a conducting solid cube of side $a$ at distance $d$ from its center. Find the electric field at the center of the cube due to the charges appearing on its surface.

1. Zero
2. $\frac{2Kq}{d^2}$
3. $\frac{Kq}{4d^2}$
4. $\frac{Kq}{d^2}$

Q. A charge $Q$ is distributed uniformly on a ring of radius $r$. A sphere of equal radius $r$ is constructed with it's centre at the periphery of the ring. Find the flux of the electric field through the surface of the sphere.

1. $\frac{Q}{{ϵ}_0}$
2. $\frac{Q}{3{ϵ}_0}$
3. $\frac{Q}{2{ϵ}_0}$
4. $\frac{Q}{4{ϵ}_0}$