Question

On charging a parallel plate capacitor to a potential V, the spacing between the plates is Halved, and a dielectric medium ${\in }_1=$ 10 is introduced between the plates, without disconnecting DC source. Explain, using suitable expression how the
1. Capacitance
2. Electric field and
3. Energy density of capacitor change

Answer 1

Hint: A parallel plate capacitor is charged to a potential V. After some time, spacing between the plates is halved and a dielectric medium of k=10 is introduced without disconnecting the DC source.

Complete Step-by-Step solution:
When the DC source remains connected, potential across the plates remains the same.
1. Initial capacitance, C=${\displaystyle \frac{{\in }_{0}A}{d}}$
When the spacing between the plate is halved d=d/2
A dielectric slab of K=10 is inserted.
New capacitance, C’=$k{\displaystyle \frac{{\in }_{0}A}{d/2}}$=10
${\displaystyle \frac{2{\in }_{0}A}{d}}=20C$
New capacitance becomes 20 times that of the initial capacitance.
2. Electric field is given by E=${\displaystyle \frac{V}{d}}$
When spacing is halved
New electric field becomes E’=${\displaystyle \frac{2V}{d}}$=2E
Therefore, the electric field becomes twice that of the initial field.
3. Initial energy density=${\displaystyle \frac{1}{2}}{\in }_0{E}^2$
                               U=${\displaystyle \frac{1}{2}}{\in }_0{E}^2$
A dielectric slab of K=10 is inserted.
New energy density U’=${\displaystyle \frac{1}{2}}K{\in }_0(2E{)}^2$
                                         $={\displaystyle \frac{1}{2}}(10){\in }_0\times 4E^2$
                                         =40$({\displaystyle \frac{1}{2}}{\in }_0{E}^2)$
                                         =40U
Therefore, energy density becomes 40 times that of the initial energy density.

Note – Here in this problem you need to know the behaviour of the capacitor when it is connected to DC supply. So, when used in a direct current or DC circuit, a capacitor charges up to its supply voltage but blocks the flow of current through it because the dielectric of a capacitor is non-conductive and basically an insulator. At this point the capacitor is said to be “fully charged” with electrons.