Six infinitely large and thin non-conducting sheets are fixed in configurations I and II. As shown in the figure, the sheets carry uniform surface charge densities which are indicated in terms of . The separation between any two consecutive sheets is . Th

Question

Six infinitely large and thin non-conducting sheets are fixed in configurations I and II. As shown in the figure, the sheets carry uniform surface charge densities which are indicated in terms of $σ_{0}$ . The separation between any two consecutive sheets is $1 μ m$ . The various regions between the sheets are denoted as 1, 2, 3, 4 and 5. If $σ_{0} = 9 μ {C/m}^{2}$ , then which of the following statements is/are correct:

(Take permittivity of free space $ε_{0} = 9 \times 10^{- 12}$ F/m) [2025]

Smart Achievers · Accepted Answer

(1)

In region 4, configuration I

${(E_{4})}_{I} = \frac{σ_{0}}{2 ε_{0}} [1 - 1 + 1 - 1 - 1 + 1] = 0$

${(V_{First})}_{I} = \frac{- σ_{0}}{2 ε_{0}} [- 1 + 2 - 3 + 4 - 5] d$

$= \frac{- σ_{0}}{2 ε_{0}} (- 3) d = \frac{3 σ_{0} d}{2 ε_{0}}$

${(V_{Last})}_{I} = \frac{- σ_{0}}{2 ε_{0}} [1 - 2 + 3 - 4 + 5] d$

$= \frac{σ_{0}}{2 ε_{0}} (- 3) d = - \frac{3 σ_{0} d}{2 ε_{0}}$

${(V_{First} - V_{Last})}_{I} = \frac{3 σ_{0} d}{ε_{0}}$

$= \frac{3 \times 9 \times 10^{- 6} \times 1 \times 10^{- 6}}{9 \times 10^{- 12}} = 3 V$

In region 3, configuration II

${(E_{3})}_{I I} = \frac{σ_{0}}{2 ε_{0}} [\frac{1}{2} - 1 + 1 + 1 - 1 + \frac{1}{2}] = \frac{σ_{0}}{2 ε_{0}}$

${(V_{First})}_{I I} = \frac{- σ_{0}}{2 ε_{0}} [- 1 + 2 - 3 + 4 - \frac{5}{2}] d$

$= \frac{- σ_{0}}{2 ε_{0}} [2 - 2.5] d = \frac{σ_{0} d}{4 ε_{0}}$

${(V_{Last})}_{I I} = \frac{- σ_{0}}{2 ε_{0}} [1 - 2 + 3 - 4 + \frac{5}{2}] d$

$= \frac{- σ_{0}}{2 ε_{0}} [6.5 - 6] d = \frac{- σ_{0} d}{4 ε_{0}}$

${(V_{First} - V_{Last})}_{I I} = \frac{σ_{0} d}{2 ε_{0}} \neq 0$

Solution

1 In region 4 of the configuration I, the magnitude of the electric field is zero.

2 In region 3 of the configuration II, the magnitude of the electric field is $\frac{σ_{0}}{ε_{0}} .$

3 Potential difference between the first and the last sheets of the configuration I is 5 V.

4 Potential difference between the first and the last sheets of the configuration II is zero.

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