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Chapter 22: Problem 50

Consider a hollow spherical conductor with total charge \(+5 e\). The outer and inner radii are \(a\) and \(b\), respectively. (a) Calculate the charge on the sphere's inner and outer surfaces if a charge of \(-3 e\) is placed at the center of the sphere. (b) What is the total net charge of the sphere?

Short Answer

Expert verified
#Answer# The charge distribution on the inner surface of the sphere is +3e, and on the outer surface is +2e. The total net charge on the sphere remains +5e.

Step by step solution

01

Apply Gauss' Law for the inner surface of the sphere

Gauss' Law states that the enclosed electric charge within a closed surface is proportional to the total electric flux through the surface. Mathematically, it's expressed as: $$\oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_\text{enclosed}}{\varepsilon_0}$$ When a charge of -3e is placed at the center of the sphere, the charge on the inner surface of the sphere \((\)radius \(b)\) should be equal and opposite to the charge placed at the center to maintain equilibrium. Thus, the charge on the inner surface of the shell, \(Q_1\), will be +3e.
02

Applying conservation of charge

The total charge on the sphere is given to be +5e. Since the charge on the inner surface is +3e, we can find the charge on the outer surface, \(Q_2\), using the conservation of charge principle: $$Q_\text{total} = Q_1 + Q_2$$ Plugging in the values: $$5e = 3e + Q_2$$
03

Solving for the charge on the outer surface

Now, just solve the equation for \(Q_2\): $$Q_2 = 5e - 3e = 2e$$ The charge on the outer surface of the sphere is +2e.
04

Calculate the total net charge on the sphere

As we have found the charge on both inner and outer surfaces of the sphere, calculating the total net charge on the sphere is straightforward. We simply sum the charge on both surfaces: $$Q_\text{net} = Q_1 + Q_2$$ Plugging in the values: $$Q_\text{net} = 3e + 2e = 5e$$ The total net charge on the sphere is +5e.

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