 # What is the total surface charge qext on the exterior surface of the conductor?

– In a conducting sphere with a neutral charge, there exists a hollow cavity inside it in the form of a sphere. There is a point charge present at the center of the spherical cavity having a positive charge $q$.

– (A) For the given conductor, find the total charge $q_{int}$ that exists on its internal surface i.e on the inside wall of the hollow sphere.

– (B) For the given conductor, find the total charge $q_{ext}$ that exists on its external surface i.e on the outside wall of the hollow sphere.

– (C) Calculate the magnitude of the electric field $\vec{E_f}$ that exists inside the hollow cavity with respect to the position of point charge $q$ assuming $k$ as follows:

$k=\frac{1}{4\pi\varepsilon_o}$

This article aims to find the impact of point charge placed in conducting sphere and the magnitude of the Electric Field generated.

Given that:

A point positive charge $q$ is placed inside a spherical cavity with radius $R$ inside a conducting sphere.

As per Gauss’s Law, the total electric flux is:

$\Phi_E=\oint\vec{E_f}\bullet\vec{ds}=\frac{q+q_{int}}{\varepsilon_o}$

Where:

$q=$ Charge at the center

$q_{int}=$ Charge induced on the inner surface

Part (A)

In a neutral sphere, the electrical field and flux are zero, hence:

$\Phi_E=\oint\vec{E_f}\bullet\vec{ds}=\frac{q+q_{int}}{\varepsilon_o}=0$

$q+q_{int}=0$

$q_{int}=-q$

Part (B)

In a neutral sphere, the net charge across its internal and external surface will be zero.

$q_{int}{+q}_{ext}=0$

$q_{ext}=-q_{int}$

$q_{ext}=-(-q)=q$

Part (C)

Let’s assume a point at a distance $r$ from the center of the cavity and $r<R$, so

$q_{enclosed}=q$

Hence:

$\Phi_E=\vec{E_f}\bullet A_{cavity}=\frac{q_{enclosed}}{\varepsilon_o}$

$\vec{E_f}=\frac{q_{enclosed}}{{A_{cavity}\times\varepsilon}_o}$

$\vec{E_f}=\frac{q}{{4\pi r^2\varepsilon}_o}=\frac{kq}{r^2}$

## Numerical Result

The total charge $q_{int}$ that exists on the inside wall of the hollow sphere is “$-q$”.

The total charge $q_{ext}$ that exists on the outside wall of the hollow sphere is “$q$”.

The magnitude of the electric field $\vec{E_f}$ inside the hollow cavity is:

$\vec{E_f}=\frac{q}{{4\pi r^2\varepsilon}_o}=\frac{kq}{r^2}$

## Example

A point charge is present at the center of the spherical cavity having a positive charge $q$. An external charge $q_2$ is brought near the conducting sphere.

(A) Find the change in the total charge $q_{int}$ on the interior surface of the cavity.

(B) Find the change in the total charge $q_{ext}$ on the exterior of the conductor.

(C) Find the change in the Electric field $E_{cav}$ inside the cavity.

(D) Find the change in the Electric field $E_{ext}$ outside the conductor.

Solution

Part (A)

The total charge $q_{int}$ is only dependent on the positive charge $q$ placed in the center, hence it will not change in the presence of external charge $q_2$.

Part (B)

The total charge $q_{ext}$ is only dependent on the positive charge $q$ placed in the center, hence it will not change in the presence of external charge $q_2$.

Part (C)

Electric field $E_{cav}$ inside the cavity will not change because the conducting sphere is neutral and the electrical field inside the cavity is only dependent on the positive charge $q$.

$\vec{E_f}=\frac{q_{enclosed}}{{A_{cavity}\times\varepsilon}_o}$

Part (D)

Electric field $E_{ext}$ outside the cavity will change because it will be distorted or affected by external charge $q_2$.

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