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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.

The basic concept behind this article is Gauss’s Law for the Electric field.

Expert Answer

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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