Concept explainers
(a)
The magnitude and the direction of the electric field for a non-
(a)
Explanation of Solution
Given:
The diameter of the sphere is
The volume charge density of the sphere is
The diameter of the shell is
The surface charge density of the sphere is
Formula used:
Write the expression of the electric field at any point for a non-conducting sphere.
Here,
Write the expression for charge for a sphere.
Here,
Substitute
Here,
Substitute
Write the above expression when
Simplify the above equation.
Write the expression for the electric field at any point due to a spherical shell.
Here,
Write the expression charge of a spherical shell.
Here,
Substitute
Here,
Substitute
Write the expression for the resultant electric field at any point in space due to a spherical shell and a solid sphere.
Calculation:
The electric field at point
Substitute
The direction of
The electric field at point
The electric field at point
Substitute
Conclusion:
Thus, the electric magnitude and the direction of electric field at point
(b)
The magnitude and the direction of the electric field for a non-conducting spherical shell concentric with a solid sphere.
(b)
Explanation of Solution
Given:
The diameter of the sphere is
The volume charge density of the sphere is
The diameter of the shell is
The surface charge density of the sphere is
Formula used:
Write the expression of the electric field at any point for a non-conducting sphere.
Here,
Write the expression for charge for a sphere.
Here,
Substitute
Here,
Substitute
Write the above expression when
Simplify the above equation.
Write the expression for the electric field at any point due to a spherical shell.
Here,
Write the expression charge of a spherical shell.
Here,
Substitute
Here,
Substitute
Resultant electric field at any point in space due to a spherical shell and a solid sphere.
Calculation:
The electric field at point
Substitute
The direction of
The electric field at point
The electric field at point
Substitute
Conclusion:
Thus, the electric magnitude and the direction of electric field at point
(c)
The magnitude and the direction of the electric field for a non-conducting spherical shell concentric with a solid sphere.
(c)
Explanation of Solution
Given:
The diameter of the sphere is
The volume charge density of the sphere is
The diameter of the shell is
The surface charge density of the sphere is
Formula used:
Write the expression of the electric field at any point for a non-conducting sphere.
Here,
Write the expression for charge for a sphere.
Here,
Substitute
Here,
Substitute
Write the above expression when
Simplify the above equation.
Write the expression for the electric field at any point due to a spherical shell.
Here,
Write the expression charge of a spherical shell.
Here,
Substitute
Here,
Substitute
Resultant electric field at any point in space due to a spherical shell and a solid sphere.
Calculation:
Write the expression for distance between the points
Write the direction for
When the above value is subtracted from
Write the expression for unit vector along
Substitute
The electric field at point
Substitute
The electric field at point
Substitute
The electric field at point
Substitute
The magnitude of electric field at
Conclusion:
Thus, the electric magnitude and the direction of electric field at point
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Chapter 22 Solutions
Physics for Scientists and Engineers
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