The ratio of the energy density between the plates for the two capacitors A and B is 1 : 4, then the ratio of the electric field intensity between the plates for the capacitor A to capacitor B is:

CONCEPT:

Capacitor:

• The capacitor is a device in which electrical energy can be stored.
  • In a capacitor two conducting plates are connected parallel to each other and carrying charges of equal magnitudes and opposite sign and separated by an insulating medium.
  • The space between the two plates can either be a vacuum or an electric insulator such as glass, paper, air, or semi-conductor called a dielectric.​

​Parallel plate capacitor:

• A parallel plate capacitor consists of two large plane parallel conducting plates separated by a small distance.
  • The space between the two plates can either be a vacuum or an electric insulator such as glass, paper, air, or semi-conductor called a dielectric.​

The energy stored in the capacitor is given as,

\(⇒ U=\frac{1}{2}QV=\frac{1}{2}CV^2=\frac{Q^2}{2C}\)

Energy density:

• It is defined as the energy stored per unit volume of space between the plates.
• Energy density u between the plates is given as,

\(⇒ u =\frac{1}{2}ϵ_oE^2\)

Where C = capacitance of the capacitor, Q = charge on the plates, V = potential difference between the plates, and E = electric field intensity between the plates

CALCULATION:

Given \(\frac{u_A}{u_B}=\frac{1}{4}\)

• We know that energy density u between the plates of a parallel plate capacitor is given as,

\(⇒ u =\frac{1}{2}ϵ_oE^2\)     -----(1)

Where E = electric field intensity between the plates, and ϵo = permittivity of vacuum

By equation 1, the energy density u_A between the plates of the capacitor A is given as,

\(⇒ u_A =\frac{1}{2}ϵ_oE_A^2\)     -----(2)

By equation 1, the energy density uB between the plates of the capacitor B is given as,

\(⇒ u_B =\frac{1}{2}ϵ_oE_B^2\)     -----(3)

By equation 2 and equation 3,

\(⇒ \frac{u_A}{u_B} =\frac{ϵ_oE_A^2}{2}\times\frac{2}{ϵ_oE_B^2}\)

\(⇒ \frac{E_A^2}{E_B^2}=\frac{1}{4}\)

\(⇒ \frac{E_A}{E_B}=\frac{1}{2}\)

• Hence, option 2 is correct.

Additional Information

Parallel plate capacitor:

• A parallel plate capacitor consists of two large plane parallel conducting plates separated by a small distance.
  • The space between the two plates can either be a vacuum or an electric insulator such as glass, paper, air, or semi-conductor called a dielectric.​
  • The electric field intensity at the outer region of the parallel plate capacitor is always zero whatever be the charge on the plate.
  • The electric field intensity in the inner region between the plates of a parallel plate capacitor remains the same at every point.
  • When the dielectric medium is filled in the space between the plates of the parallel plate capacitor, its capacitance increases.
• The electric field intensity in the inner region between the plates of a parallel plate capacitor is given as,

• The potential difference between the plates is given as,

• The capacitance C of the parallel plate capacitor is given as,

​Where A = area of the plates, d = distance between the plates, Q = charge on the plates, σ = surface charge density, E = electric field between the plates, and K = dielectric constant