SATHEE: Electrostatics 3 Question 19

Electrostatics 3 Question 19

19. Three concentric spherical metallic shells, $A, B$ and $C$ of radii $a, b$ and $c (a < b < c)$ have surface charge densities $σ, - σ$ and $σ$ respectively.

(1990, 7M)

(a) Find the potential of the three shells $A, B$ and $C$ .

(b) If the shells $A$ and $C$ are at the same potential, obtain the relation between the radii $a, b$ and $c$ .

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Answer:

Correct Answer: 19. $\begin{aligned} (a) V_{A} = \frac{σ}{ε_{0}} (a - b + c), V_{B} = \frac{σ}{ε_{0}} (\frac{a^{2}}{b} - b + c), & V_{C} = \frac{σ}{ε_{0}} (\frac{a^{2}}{c} - \frac{b^{2}}{c} + c) & (b) a + b = c \end{aligned}$

Solution:

(a) Potential at any shell will be due to all three charges.

$\begin{aligned} V_{A} & = \frac{1}{4 π ε_{0}} \frac{q_{A}}{a} + \frac{q_{B}}{b} + \frac{q_{C}}{c} \\ = \frac{1}{4 π ε_{0}} \frac{(4 π a^{2}) (σ)}{a} + \frac{(4 π b^{2}) (- σ)}{b} + \frac{(4 π c^{2}) (σ)}{c} \\ = \frac{σ}{ε_{0}} (a - b + c) \\ V_{B} & = \frac{1}{4 π ε_{0}} \frac{q_{A}}{b} + \frac{q_{B}}{b} + \frac{q_{C}}{c} \\ = \frac{1}{4 π ε_{0}} \frac{(4 π a^{2}) (σ)}{b} + \frac{(4 π b^{2}) (- σ)}{b} + \frac{(4 π c^{2}) (σ)}{c} \\ = \frac{σ}{ε_{0}} \frac{a^{2}}{b} - b + c \end{aligned}$

Similarly, $V_{C} = \frac{1}{4 π ε_{0}} \frac{q_{A}}{c} + \frac{q_{B}}{c} + \frac{q_{C}}{c}$

$\begin{aligned} = \frac{1}{4 π ε_{0}} \frac{(4 π a^{2}) (σ)}{c} + \frac{(4 π b^{2}) (- σ)}{c} + \frac{(4 π c^{2}) (σ)}{c} \\ = \frac{σ}{ε_{0}} \frac{a^{2}}{c} - \frac{b^{2}}{c} + c \end{aligned}$

(b) Given $V_{A} = V_{C}$

$\begin{aligned} ∴ \frac{σ}{ε_{0}} (a - b + c) = \frac{σ}{ε_{0}} \frac{a^{2}}{c} - \frac{b^{2}}{c} + c \\ ∴ a - b + c = \frac{a^{2}}{c} - \frac{b^{2}}{c} + c \\ or a + b = c \end{aligned}$

Electrostatics 3 Question 19

19. Three concentric spherical metallic shells, $A, B$ and $C$ of radii $a, b$ and $c (a < b < c)$ have surface charge densities $σ, - σ$ and $σ$ respectively.

Answer:

Solution:

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Electrostatics 3 Question 19

19. Three concentric spherical metallic shells, A,B and C of radii a,b and c(a<b<c) have surface charge densities σ,−σ and σ respectively.

Answer:

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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19. Three concentric spherical metallic shells, $A, B$ and $C$ of radii $a, b$ and $c (a < b < c)$ have surface charge densities $σ, - σ$ and $σ$ respectively.