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Chapter 11: Problem 1575

Three concentric spherical shells have radii a, \(b\) and \(c(a

Short Answer

Expert verified
(D) \(\mathrm{V}_{\mathrm{C}}=\mathrm{V}_{\mathrm{A}} \neq \mathrm{V}_{\mathrm{B}}\)

Step by step solution

01

Write the expression for potential

For a spherical shell with a radius r and surface charge density σ, the potential V can be written as: \[V = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r}\] where Q is the total charge on the shell, and \(\epsilon_0\) is the vacuum permittivity. For our problem, we will denote the total charge for each shell as \(Q_A\), \(Q_B\), and \(Q_C\).
02

Find the total charge on each shell

The total charge on a spherical shell is given by: \[Q = \sigma \times \text{Surface Area}\] For each shell, their total charges are: - \(Q_A = \sigma \times 4\pi a^2\) - \(Q_B = -\sigma \times 4\pi b^2\) - \(Q_C = \sigma \times 4\pi (a+b)^2\)
03

Calculate the potentials for each shell

Now, we can substitute the corresponding total charges into the expression for potential to calculate the potentials for each shell: - \(V_A = \frac{1}{4 \pi \epsilon_0} \frac{\sigma \times 4\pi a^2}{a} = \frac{\sigma a}{\epsilon_0}\) - \(V_B = \frac{1}{4 \pi \epsilon_0} \frac{-\sigma \times 4\pi b^2}{b} = -\frac{\sigma b}{\epsilon_0}\) - \(V_C = \frac{1}{4 \pi \epsilon_0} \frac{\sigma \times 4\pi (a+b)^2}{(a + b)} = \frac{\sigma (a+b)}{\epsilon_0}\)
04

Compare the potentials

Now, we can compare the potentials of the shells: - \(V_C = \frac{\sigma (a+b)}{\epsilon_0} = \frac{\sigma a}{\epsilon_0} + \frac{\sigma b}{\epsilon_0} = V_A - V_B\) From this result, we can see that \(V_C = V_A \neq V_B\), which corresponds to option (D). Therefore, the correct answer is: (D) \(\mathrm{V}_{\mathrm{C}}=\mathrm{V}_{\mathrm{A}} \neq \mathrm{V}_{\mathrm{B}}\)

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Most popular questions from this chapter

The plates of a parallel capacitor are charged up to \(100 \mathrm{~V}\). If $2 \mathrm{~mm}$ thick plate is inserted between the plates, then to maintain the same potential difference, the distance between the capacitor plates is increased by \(1.6 \mathrm{~mm}\) the dielectric constant of the plate is (A) 5 (B) 4 (C) \(1.25\) (D) \(2.5\)

Two electric charges \(12 \mu \mathrm{c}\) and \(-6 \mu \mathrm{c}\) are placed \(20 \mathrm{~cm}\) apart in air. There will be a point \(P\) on the line joining these charges and outside the region between them, at which the electric potential is zero. The distance of \(P\) from \(-6 \mu c\) charge is ..... (A) \(0.20 \mathrm{~m}\) (B) \(0.10 \mathrm{~m}\) (C) \(0.25 \mathrm{~m}\) (D) \(0.15 \mathrm{~m}\)

The electric potential \(\mathrm{V}\) at any point $\mathrm{x}, \mathrm{y}, \mathrm{z}\( (all in meter) in space is given by \)\mathrm{V}=4 \mathrm{x}^{2}$ volt. The electric field at the point \((1 \mathrm{~m}, 0,2 \mathrm{~m})\) in \(\mathrm{Vm}^{-1}\) is \((\mathrm{A})+8 \mathrm{i} \wedge\) (B) \(-8 \mathrm{i} \wedge\) (C) \(-16 \mathrm{i}\) (D) \(+16 \mathrm{i}\)

A parallel plate capacitor with air between the plates has a capacitance of $9 \mathrm{pF}\(. The separation between its plates is \)\mathrm{d}$. The space between the plates is now filled with two dielectrics. One of the dielectric constant \(\mathrm{K}_{1}=3\) and thickness \(\mathrm{d} / 3\) while the other one has dielectric constant \(\mathrm{K}_{2}=6\) and thickness \(2 \mathrm{~d} / 3\). Capacitance of the capacitor is now (A) \(1.8 \mathrm{pF}\) (B) \(20.25 \mathrm{pF}\) (C) \(40.5 \mathrm{pF}\) (D) \(45 \mathrm{pF}\)

Two identical capacitors have the same capacitance \(\mathrm{C}\). one of them is charged to a potential \(\mathrm{V}_{1}\) and the other to \(\mathrm{V}_{2}\). The negative ends of the capacitors are connected together. When the positive ends are also connected, the decrease in energy of the combined system is (A) \((1 / 4) \mathrm{C}\left(\mathrm{V}_{1}^{2}-\mathrm{V}_{2}^{2}\right)\) (B) \((1 / 4) \mathrm{C}\left(\mathrm{V}_{1}^{2}+\mathrm{V}_{2}^{2}\right)\) (C) \((1 / 4) \mathrm{C}\left(\mathrm{V}_{1}-\mathrm{V}_{2}\right)^{2}\) (D) \((1 / 4) \mathrm{C}\left(\mathrm{V}_{1}+\mathrm{V}_{2}\right)^{2}\)
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