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Charge Induction in Metal Cavities

Charge Induction in Metal Cavities :- We know that static charges cannot produce an electric field inside a conductor. Hence, no electric field lines can enter a conducting body.

Consider a point charge +q placed at the center of a spherical cavity within a metal. The total electric flux from +q is confined by an induced charge –q on the inner surface of the cavity, ensuring no field enters the surrounding metal. So all the electric flux starting from point charge +q ends at the induced charge -q on the inner surface of the cavity.

At any point A inside the metal, the net electric field is zero. This means the field due to +q is exactly canceled by the field from the induced charges on the inner surface of cavity. The outer surface of the conductor gains a +q charge, which distributes itself in such a way that it does not create any electric field within the conductor.

From the above analysis, we can conclude that if a charge is placed in the cavity of a metal body :

(i). An equal and opposite charge is induced on the inner surface of the cavity.

(ii). A similar and equal amount of charge is induced on the outer surface of the body with surface charge density inversely proportional to the radius of curvature of the body.

(iii). If the charge inside the cavity is displaced, the induced charge distribution on the inner surface of the cavity adjusts in such a way that its center of charge effectively follows the point charge, thereby ensuring the electric field in the outer region remains zero.

(iv). The movement of the point charge inside the metal cavity of the conductor does not affect the charge distribution on the outer surface, as illustrated above.

(v). If some other charge is brought near the body from outside, it will only affect the charge distribution on the outer surface, not the distribution inside the cavity, as shown in the figure below.

Finding Electric Field Intensity and Electric Potential in case of

Charge Induction in Metal Cavities

(a). A Conductor Having a Spherical Cavity with a Charge At The Centre

Consider a spherical conductor of radius with a spherical cavity of radius () having a point charge placed at the center. As a result of induction, a charge of –q is induced on the inner surface of the cavity, and an equal charge of +q is induced on the outer surface of the conductor.

Let us find the values of electric field and electric potential at three points :

Inside the cavity
Inside the metal
Outside the conductor

(1). Inside the cavity ( )

The electric potential at a distance r from the center within the cavity :

$\displaystyle {{V}_{centre}}={{\left[ \frac{kq}{r} \right]}_{due\text{ }to\text{ }q}}+{{\left[ \frac{k\left( -q \right)}{{{R}_{1}}} \right]}_{due\text{ }to\text{ }charge\text{ }on\text{ }inner\text{ }surface\text{ }of\text{ }cavity}}+{{\left[ \frac{k\left( +q \right)}{{{R}_{2}}} \right]}_{due\text{ }to\text{ }charge\text{ }on\text{ }outer\text{ }surface\text{ }of\text{ }conductor}}$

$\displaystyle \Rightarrow {{V}_{centre}}=kq\left[ \frac{1}{r}-\frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}} \right]$

The electric field intensity at a distance r from the center within the cavity :

$\displaystyle {{E}_{r<{{R}_{1}}}}={{\left[ \frac{kq}{{{r}^{2}}} \right]}_{only\text{ }due\text{ }to\text{ }q}}$

(2). Inside the metal ( $< R_{2}$ )

Electric potential inside the metal at a distance r from the centre :

$\displaystyle {{V}_{{{R}_{1}}<r<{{R}_{2}}}}={{\left[ \frac{kq}{r} \right]}_{due\text{ }to\text{ }q}}+{{\left[ \frac{k\left( -q \right)}{r} \right]}_{due\text{ }to\text{ }charge\text{ }on\text{ }inner\text{ }surface\text{ }of\text{ }cavity}}+{{\left[ \frac{k\left( +q \right)}{{{R}_{2}}} \right]}_{due\text{ }to\text{ }charge\text{ }on\text{ }outer\text{ }surface\text{ }of\text{ }conductor}}$

$\displaystyle \Rightarrow {{V}_{{{R}_{1}}<r<{{R}_{2}}}}=\frac{kq}{{{R}_{2}}}$

Electric field intensity inside the metal at a distance r from the centre :

$\displaystyle {{E}_{{{R}_{1}}<r<{{R}_{2}}}}=0$

This is because inside a conductor, electric field is zero.

(3). Outside the conductor ( )

Electric potential outside the conductor at a distance r from the centre :

$\displaystyle {{V}_{r>{{R}_{2}}}}={{\left[ \frac{kq}{r} \right]}_{due\text{ }to\text{ }q}}+{{\left[ \frac{k\left( -q \right)}{r} \right]}_{due\text{ }to\text{ }charge\text{ }on\text{ }inner\text{ }surface\text{ }of\text{ }cavity}}+{{\left[ \frac{k\left( +q \right)}{r} \right]}_{due\text{ }to\text{ }charge\text{ }on\text{ }outer\text{ }surface\text{ }of\text{ }conductor}}$

$\displaystyle \Rightarrow {{V}_{r>{{R}_{2}}}}=\frac{kq}{r}$

Electric field intensity outside the conductor at a distance r from the centre :

$\displaystyle {{{\vec{E}}}_{r>{{R}_{2}}}}={{\left[ \frac{kq}{{{r}^{2}}}\hat{r} \right]}_{due\text{ }to\text{ }q}}+{{\left[ \frac{k\left( q \right)}{{{r}^{2}}}(-\hat{r}) \right]}_{due\text{ }to\text{ }charge\text{ }on\text{ }inner\text{ }surface\text{ }of\text{ }cavity}}+{{\left[ \frac{kq}{{{r}^{2}}}\hat{r} \right]}_{due\text{ }to\text{ }charge\text{ }on\text{ }outer\text{ }surface\text{ }of\text{ }conductor}}$

$\displaystyle \Rightarrow {{E}_{r>{{R}_{2}}}}=\frac{kq}{{{r}^{2}}}$

(b). A Conductor Having a Spherical Cavity containing a Charge Slightly Off-centre

Suppose a point charge +q is placed at a distance r from the center of the cavity. The induced charges on different surfaces are as shown in the following figure :

We will find the values of electric field and electric potential at three points :

Inside the cavity
Inside the metal
Outside the conductor

(1). At the centre of the cavity

Electric potential at the centre of the cavity :

$\displaystyle \Rightarrow {{V}_{centre}}=\frac{kq}{r}-\frac{kq}{{{R}_{1}}}+\frac{kq}{{{R}_{2}}}$

Electric field intensity at the centre of the cavity (here, only the contribution of the point charge +q is considered) :

$\displaystyle {{E}_{centre}}={{\left[ \frac{kq}{{{r}^{2}}} \right]}_{only\text{ }due\text{ }to\text{ }q}}$

(2). Inside the metal

Electric potential at a point inside the metal due to the system :

The point charge +q is not placed at the center but slightly off-center, which breaks the spherical symmetry of the system. As a result, the electric potential inside the conductor now also depends on direction. There is no general solution for this situation; the solution depends on the specific details provided in the problem.

Electric field intensity at a point inside the metal due to the system :

$\displaystyle E=0$

Because the electric field inside a conductor is zero.

(3). At a point outside the conductor

If we find the electric field intensity and electric potential at a point located at a distance r from the center, outside the system, they will be solely due to the induced charge on the outer surface of the conductor. This is because the induced charge on the inner surface of the cavity cancels the effect of the point charge placed inside the cavity.

Electric potential at a point outside the conductor, located at a distance r from its center :

$\displaystyle \Rightarrow {{V}_{r>{{R}_{2}}}}=\frac{kq}{r}-\frac{kq}{r}+\frac{kq}{r}$

$\displaystyle \therefore {{V}_{r>{{R}_{2}}}}=\frac{kq}{r}$

Electric field intensity at a point outside the conductor, located at a distance r from its center :

$\displaystyle {{E}_{r>{{R}_{2}}}}=\frac{kq}{{{r}^{2}}}$

(c). A Conductor with a Spherical Cavity and Charge Given to Its Outer Surface

When a charge is given to a conductor, it always resides on its outer surface in electrostatic equilibrium. Let us find the values of electric field and electric potential at three points :

Inside the cavity
Inside the metal
Outside the conductor

(1). Inside the cavity ( )

The electric potential at centre due to the system :

$\displaystyle {{V}_{centre}}={{\left[ \frac{k\left( +q \right)}{{{R}_{2}}} \right]}_{due\text{ }to\text{ }charge\text{ }on\text{ }outer\text{ }surface}}=\frac{kq}{{{R}_{2}}}$

The electric field intensity at centre :

$\displaystyle {{E}_{centre}}=0$

Because no charge is enclosed by the cavity.

(2). Inside the metal ( $< R_{2}$ )

Electric potential :

$\displaystyle {{V}_{centre}}={{\left[ \frac{k\left( +q \right)}{{{R}_{2}}} \right]}_{due\text{ }to\text{ }charge\text{ }on\text{ }outer\text{ }surface}}=\frac{kq}{{{R}_{2}}}$

Electric field intensity :

$\displaystyle {{E}_{{{R}_{1}}<r<{{R}_{2}}}}=0$

Because no charge is enclosed by a sphere of radius r ( $< R_{2}$ ).

(3). Outside the conductor ( )

Electric potential :

$\displaystyle {{V}_{centre}}={{\left[ \frac{k\left( +q \right)}{r} \right]}_{due\text{ }to\text{ }charge\text{ }on\text{ }outer\text{ }surface}}=\frac{kq}{r}$

Electric field intensity :

$\displaystyle {{E}_{r>{{R}_{2}}}}=\frac{kq}{{{r}^{2}}}$

Charge Induction in Metal Cavities

Finding Electric Field Intensity and Electric Potential in case of

Charge Induction in Metal Cavities

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