Concept explainers
(a)
The total and individual charge on the inner and outer surface of the spherical shell.
(a)
Explanation of Solution
Given:
A positive point charge of
Formula used:
The positive charge at the center of the spherical sphere induces a charge of same magnitude but different in polarity on the inner surface. The negative charge on the inner surface induces a positive charge of same magnitude on the outer surface.
Write the expression for the surface charge density
Here,
Write the expression for the surface area of the square sheet.
Here
Substitute
Forinner surface of the spherical shell.
The induced charge on the inner surface has same magnitude of the point charge .The polarity of the charge is opposite to the point charge.
Calculation:
Substitute
For outer surface of the spherical shell
The induced charge on the outer surface has same magnitude of the point charge .The polarity of the charge is same to the point charge
Substitute
Conclusion:
Thus, the total charge on the inner surface is
(b)
The electric field due to a positive point charge located at the spherical shell.
(b)
Explanation of Solution
Given:
A positive point charge of
Formula used:
Write the expression for Gauss’s law.
Here,
Substitute
Here
Rearrange the above equation for
Substitute
Calculation:
For
Substitute
For
Substitute
For
Substitute
Conclusion:
Thus, the electric field of the spherical shell for
(c)
The electric field and surface charge density of the spherical shell.
(c)
Explanation of Solution
Given:
A positive point charge of
Formula used:
The positive charge at the center of the spherical sphere induces a charge of same magnitude but different in polarity on the inner surface. The negative charge on the inner surface induces a positive charge of same magnitude on the outer surface.
Write the expression for the surface charge density
Here,
Write the expression for the surface area of the square sheet.
Here
Substitute
For inner surface of the spherical shell.
The induced charge on the inner surface has same magnitude of the point charge .The polarity of the charge is opposite to the point charge
Write the expression for Gauss’s law.
Here,
Substitute
Here
Rearrange the above equation for
Substitute
Calculation:
For inner surface
Substitute for
For outer surface of the spherical shell
Write the expression for surface charge.
Substitute
For outer surface.
Substitute for
For
Substitute
For
Substitute
For
Substitute
Conclusion:
Thus, the total charge on the inner surface is
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Chapter 22 Solutions
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