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GATE 2021 PH – Question 41

Electromagnetic theory · Dielectrics and conductors · 2 marks · Multiple choice

Two concentric conducting spherical shells have radii $a,b$ and charges $+Q,-Q$, respectively. Half the space between them is filled with a hemispherical dielectric of permittivity $\epsilon_1$; the other half contains air of permittivity $\epsilon_0$. For $a<r<b$, the electric field is

Diagram for GATE 2021 PH question 41
  1. $\frac{Q\hat{\mathbf r}}{2\pi(\epsilon_0+\epsilon_1)r^2}$ everywhere
  2. $\frac{Q\hat{\mathbf r}}{4\pi\epsilon_0r^2}$ in air and $\frac{Q\hat{\mathbf r}}{4\pi\epsilon_1r^2}$ in dielectric
  3. $\frac{Q\hat{\mathbf r}}{2\pi\epsilon_0r^2}$ in air and $\frac{Q\hat{\mathbf r}}{2\pi\epsilon_1r^2}$ in dielectric
  4. $\frac{Q\hat{\mathbf r}}{4\pi(\epsilon_0+\epsilon_1)r^2}$ everywhere

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Correct answer: (A) $\frac{Q\hat{\mathbf r}}{2\pi(\epsilon_0+\epsilon_1)r^2}$ everywhere

Explanation

The dielectric interface is radial, so a radial electric field is tangential to it and must be continuous. Thus the field has the same magnitude in both hemispheres. Apply Gauss’s law to displacement: $Q=2\pi r^2\epsilon_0E+2\pi r^2\epsilon_1E$. Solving gives $E=Q/[2\pi(\epsilon_0+\epsilon_1)r^2]$, directed radially outward. The free charge redistributes between the hemispheres; it is not divided equally in general.

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