Three concentric spherical metallic shells $\mathrm{X}$, $\mathrm{Y}$ and $\mathrm{Z}$ of radius $\mathrm{a}$, $\mathrm{b}$ and $\mathrm{c}$ respectively $[\mathrm{a}<\mathrm{b}<\mathrm{c}]$ have surface charge densities $\sigma$, $-\sigma$ and $\sigma$, respectively. The shells X and Z are at same potential. If the radii of $X \& Y$ are 2 cm and 3 cm, respectively. The radius of shell Z is $\_\_\_\_$ cm.

JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Electrostatics

Three concentric spherical metallic shells

X

Y

and

Z

of radius

a

b

and

c

respectively

[a < b < c]

have surface charge densities

σ

- σ

and

σ

, respectively. The shells X and Z are at same potential. If the radii of

X & Y

are 2 cm and 3 cm, respectively. The radius of shell Z is

____

cm.

Answer: 5

Each shell carries

q = σ (4 π r^{2})

q_{X} = 4 π a^{2} σ

q_{Y} = - 4 π b^{2} σ

q_{Z} = 4 π c^{2} σ

. The potential of the innermost shell is

V_{X} = 4 π k σ (a - b + c)

, while the outermost shell, enclosing all the charge, has

V_{Z} = \frac{4 π k σ ( a ^{2} - b ^{2} + c ^{2} )}{c}

. Setting

V_{X} = V_{Z}

gives

c (a - b) = a^{2} - b^{2} = (a - b) (a + b)

, so

c = a + b = 2 + 3 = 5

. The radius of shell Z is

5

cm.

JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Integer answer