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Chapter 21: Problem 30

A \(10- \mathrm{nC}\) point charge is located at the center of a thin spherical shell of radius \(8.0 \mathrm{cm}\) carrying \(-20 \mathrm{nC}\) distributed uniformly over its surface. Find the magnitude and direction of the electric field (a) \(2.0 \mathrm{cm},\) (b) \(6.0 \mathrm{cm},\) and (c) \(15 \mathrm{cm}\) from the point charge.

Short Answer

Expert verified
The magnitude and direction of the electric field at each point are as follows: (a) 2.25 * 10^7 N/C, directed outward. (b) 2.50 * 10^6 N/C, directed outward. (c) 1.00 * 10^6 N/C, directed inward.

Step by step solution

01

Calculation of Electric Field due to Point Charge

The electric field \(E\) due to a point charge \(Q\) at a distance \(r\) is given by Coulomb's law: \(E = k \cdot \frac{Q}{r^2}\), where \(k = 9*10^9 Nm^2C^-2\) is Coulomb's constant. Calculate the same for each point (2cm, 6cm, 15cm).
02

Calculation of Electric Field due to Spherical Shell

When dealing with a uniformly charged thin spherical shell, the field inside the shell is always zero due to symmetry regardless of the amount of charge. Outside the shell however, the shell behaves as if all its charge is concentrated at its center. Thus, the electric field for a point outside the shell (15cm) can be calculated using Coulomb's law.
03

Superposition of Fields

For each given point, the net electric field is the vector sum of the individual electric fields due to the point charge and the shell. Specifically, for point (a), electric field will only be due to the point charge, for point (b), it will still only be due to the point charge, and for point (c), we need to calculate the superposition of electric fields due to both the charge and the shell.

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Most popular questions from this chapter

An electric field is given by \(\vec{E}=E_{0}(y / a) \hat{k},\) where \(E_{0}\) and \(a\) are constants. Find the flux through the square in the \(x\) -y plane bounded by the points \((0,0),(0, a),(a, a),(a, 0)\).

The electric flux through a closed surface is zero. Must the electric field be zero on that surface? If not, give an example.

Charges \(+2 q\) and \(-q\) are near each other. Sketch some field lines for this charge distribution, using eight lines for a charge of magnitude \(q\).

A point charge \(q\) is at the center of a spherical shell of radius \(R\) carrying charge \(2 q\) spread uniformly over its surface. Write expressions for the electric field strength at (a) \(\frac{1}{2} R\) and (b) \(2 R\).

The electric field of a flat sheet of charge is \(\sigma / 2 \epsilon_{0} .\) Yet the field of a flat conducting sheet-even a thin one, like a piece of aluminum foil- is \(\sigma / \epsilon_{0} .\) Explain this apparent discrepancy.
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