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## Electric Field from Electric Potential

If the electric potential is given by (\chi=cxy), calculate the electric field.

Discussion:

$$E_x=-\frac{\partial\chi}{\partial x}=-cy$$
$$E_y=\frac{\partial \chi}{\partial y}=-cx$$
Hence electric field $$\vec{E}=-c(y\hat{i}+x\hat{j})$$

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## Three Charged Spheres

Let’s discuss a problem where we will find the charge of the sphere with the help of the three charged spheres problem. Try before reading the solution.

The Problem:

Consider three concentric metallic spheres $A$, $B$ and $C$ of radii of $a$, $b$, $c$ respectively where a<b<c . $A$ and $B$ are connected whereas C is grounded The potential of the middle sphere $B$ is raised to $V$ then what is the charge on the sphere $C$?

Solution:

Three concentric metallic spheres $A$, $B$ and $C$ have radii of $a$, $b$, $c$ respectively where a<b<c . $A$ and $B$ are connected whereas C is grounded The potential of the middle sphere $B$ is raised to $V$.

$$V=\frac{Kq}{b}+\frac{KQ}{c}$$ $$\Rightarrow \frac{k(q+Q)}{c}=0$$
$$\Rightarrow q+Q=0$$
$$\Rightarrow q=-Q$$
$$\frac{k(-Q)}{b}+\frac{KQ}{c}=V$$
$$\Rightarrow KQ(\frac{1}{c}-\frac{1}{b})=V$$
$$Q=\frac{bcV}{(b-c)}4\pi\epsilon_0$$

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## Potential of Metal Sphere

Let’s discuss the problem where we have to find the potential of metal sphere.

The Problem:

A metal sphere having a radius (r_1) charged to a potential (\phi_1) is enveloped by a thin-walled conducting spherical shell of radius (r_2). Determine the potential (\phi_2) acquired by the sphere after it has been connected for a short time to the shell by a conductor.

Solution:

The charge (q_1) of the sphere can be determined from the relation $$q_1=4\pi\epsilon_0r_1$$
After the connection of the sphere to the envelope, the entire charge (q_1) will flow from the sphere to the envelope and will be distributed uniformly over its surface.
Its potential (\phi_2) (coinciding with the new value of the potential of the sphere) will be
$$\phi_2=\frac{q_1}{4\pi\epsilon_0r_2}=\phi_1\frac{r_1}{r_2}$$