Question

A sheet of aluminum foil of negligible thickness is introduced between the plates of a capacitor. The capacitance of the capacitor:
A. Decreases
B. Remains unchanged
C. Becomes infinite
D. Increases

Answer 1

Hint: After the introduction of the aluminum foil of negligible thickness the plates will behave as two capacitors being connected in series. Then find the effective capacitance of this system. Compare the capacitances of both the systems and choose options accordingly.

Complete step by step answer:
As per the given condition, we have that the aluminum foil of negligible thickness is inserted in the middle of the plates of a capacitor. Therefore, the system will now act as two capacitors connected in series.
The capacitance of parallel plate is given as:
$C = \dfrac{{{\varepsilon _0}A}}{d}$
Let this capacitance be $\dfrac{{{\varepsilon _0}A}}{d} = x$ --equation $1$
Here ${\varepsilon _0}$ is the permittivity of free space, A is the area of the capacitor plate and d is the distance between the two plates.
Let us calculate the capacitance of the two plates in series combination as follows:
Let the capacitance of the first capacitor be ${C_1}$ and the capacitance of the second capacitor be ${C_2}$ .The new distance between the capacitor plates and aluminium foil is ${d_0} = \dfrac{d}{2}$ .
$ \Rightarrow {C_1} = {C_2} = \dfrac{{{\varepsilon _0}A}}{{{d_0}}}$
$ \Rightarrow {C_1} = {C_2} = \dfrac{{{\varepsilon _0}A}}{{\dfrac{d}{2}}}$
$ \Rightarrow {C_1} = {C_2} = \dfrac{{2{\varepsilon _0}A}}{d}$
From equation $1$ we have
$ \Rightarrow {C_1} = {C_2} = 2x$ --equation $2$
Now, the effective capacitance ${C_{eff}}$ in series combination is given as:
$\dfrac{1}{{{C_{eff}}}} = \dfrac{1}{{{C_1}}} + \dfrac{1}{{{C_2}}}$
But from equation $2$ we can have
$\dfrac{1}{{{C_{eff}}}} = \dfrac{1}{{2x}} + \dfrac{1}{{2x}}$
$ \Rightarrow \dfrac{1}{{{C_{eff}}}} = \dfrac{1}{x}$
$ \Rightarrow {C_{eff}} = x$
$ \therefore{C_{eff}} = \dfrac{{{\varepsilon _0}A}}{d}$
We observe that this value is equal to the capacitance when aluminium foil was not introduced. This means that in both the cases, the value of capacitances is the same.The capacitance of the capacitor after a sheet of aluminum foil of negligible thickness is introduced between the plates remains unchanged.

Thus, B is the correct option.

Note:When we insert a foil of negligible thickness between the parallel plates of the capacitor, the area of the plates does not change. As the medium in which the capacitor is placed is not given, we considered the medium to be vacuum else the permittivity of the free space will also be changed.