Concept explainers
A solid, insulating sphere of radius a has a uniform charge density throughout its volume and a total charge Q. Concentric with this sphere is an uncharged,
(a)
The charge enclosed by the Gaussian surface in the region
Answer to Problem 75P
The charge enclosed by the Gaussian surface in the region
Explanation of Solution
Figure 1 represents an insulating sphere of radius
Consider a Gaussian surface of radius
Write the expression for charge enclosed by the Gaussian surface in the region
Here,
Write the expression for volume enclosed by the Gaussian surface.
Here,
Use equation (II) in (I).
Write the expression for volume charge density for the insulating sphere of radius
Use equation (IV) in (III),to find
Conclusion:
Therefore, the charge enclosed by the Gaussian surface in the region
(b)
The electric field in the region
Answer to Problem 75P
The electric field in the region
Explanation of Solution
Write the expression for Gauss law.
Here,
Due to spherical symmetry, the element of area
From subpart (a), the charge enclosed in the region
Use equation (V) in (VII), and rearrange.
The above is written as
Here,
Conclusion:
Therefore, the electric field in the region
(c)
The charge enclosed by the Gaussian surface in the region
Answer to Problem 75P
The charge enclosed by the Gaussian surface in the region
Explanation of Solution
Write the expression for charge enclosed by the Gaussian surface in the region
Here,
Conclusion:
Therefore, charge enclosed by the Gaussian surface in the region
(d)
The magnitude of electric field in the region
Answer to Problem 75P
The magnitude of electric field in the region
Explanation of Solution
Write the expression for charge enclosed by the Gaussian surface in the region
Here,
Use equation (XI) in (VII).
The above equation is written as
Conclusion:
Therefore, the magnitude of electric field in the region
(e)
The magnitude of electric field in the region
Answer to Problem 75P
The magnitude of electric field in the region
Explanation of Solution
Write the expression for Gauss law.
The Gaussian surface encloses zero charge in the region
Write the expression for charge enclosed by the Gaussian surface in the region
Here,
Use equation (XIII) in (VI).
Conclusion:
Therefore, the magnitude of electric field in the region
(f)
The charge on the inner surface of the hollow sphere.
Answer to Problem 75P
The charge on the inner surface of the hollow sphere is
Explanation of Solution
Write the expression for Gauss law.
Write the expression for charge enclosed by the Gaussian surface.
Here,
Use equation (XIV) in (VI).
The electric field inside the conductor is zero, the charge enclosed by the Gaussian surface is zero. The above equation is reduced to
Conclusion:
Therefore, the charge on the inner surface of the hollow sphere is
(g)
The charge on the outer surface of the hollow sphere.
Answer to Problem 75P
The charge on the outer surface of the hollow sphere is
Explanation of Solution
Write the expression for total charge inside the hollow sphere.
Here,
The total charge inside the conductor is zero, the above equation is reduced to
Rearrange the above equation, to find
Conclusion:
Substitute
Therefore, The charge on the outer surface of the hollow sphere is
(h)
Among the three spherical surfaces having radii
Answer to Problem 75P
The inner surface of radius
Explanation of Solution
Write the expression for surface charge density.
Here,
The solid insulating sphere has small surface charge density since its total charge is uniformly distributed throughout its volume.
The inner surface of the hollow cylinder with radius
Conclusion:
Therefore, the inner surface of radius
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Chapter 19 Solutions
Principles of Physics: A Calculus-Based Text
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