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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 body 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). Two Concentric Spherical Metallic Shells with a Charged Particle Placed Inside the Smaller Shell

Consider two concentric metallic shells of radii and (). Let a point charge be placed at a distance from the center, inside the smaller shell.

(1). At the centre of the shells

Electric potential at centre due to the system :

$\displaystyle {{V}_{centre}}={{\left[ \frac{kq}{r} \right]}_{due\text{ }to\text{ }q}}+{{\left[ \frac{k\left( -q \right)}{{{R}_{1}}} \right]}_{due\text{ }to\text{ }Inner\text{ }shell}}+{{\left[ \frac{k\left( +q \right)}{{{R}_{2}}} \right]}_{due\text{ }to\text{ }Outer\text{ }shell}}$

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

Electric field intensity at centre due to the system (Only the point charge contributes here) :

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

(2). Outside the outer shell

If we find the electric field intensity and electric potential at a distance from the center, outside the shells, they will be due only to the charge induced on the outer shell. This is because the charge induced on the inner surface of the smaller shell nullifies the effect of the point charge inside it.

Electric potential outside the outer shell at a distance r from the centre due to the system :

$\displaystyle {{V}_{r>{{R}_{2}}}}={{\left[ \frac{kq}{r} \right]}_{due\text{ }to\text{ }q}}+{{\left[ \frac{k\left( -q \right)}{r} \right]}_{due\text{ }to\text{ }Inner\text{ }shell}}+{{\left[ \frac{k\left( +q \right)}{r} \right]}_{due\text{ }to\text{ }Outer\text{ }shell}}$

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

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

Electric field intensity outside the outer shell at a distance r from the centre due to the system :

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

(b). 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 :

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

(c). A Conductor Having a Spherical Cavity with a Charge Inside It

Now consider the case in which a charge +q is placed at the center of the cavity. As a result, a charge –q is induced on the inner surface of the cavity, and an equal charge +q appears 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 centre inside the cavity :

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

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

The electric field intensity at a distance r from centre inside 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 at a distance r from centre :

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

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

Electric field intensity at a distance r from 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

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

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

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

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