Concept explainers
Review. Consider a capacitor with vacuum between its large, closely spaced, oppositely charged parallel plates. (a) Show that the force on one plate can be accounted for by thinking of the electric field between the plates as exerting a “negative pressure” equal to the energy density of the electric field. (b) Consider two infinite plane sheets carrying
(a)
To show: The force on one plate can be accounted for by thinking of the electrical field between the plates as exerting a negative pressure equal to the energy density of the electric field.
Answer to Problem 32.60AP
Explanation of Solution
The formula to force between the plates of capacitor is,
Here,
The formula to calculate electric field produced by negative capacitor plate is,
The formula to calculate electric field produced by positive capacitor plate is,
The formula to calculate net electric field produced between the plates is,
Substitute
The energy density of capacitor is,
Substitute
Rearrange above equation.
Here,
Hence, the force on one plate can be accounted for by thinking of the electrical field between the plates as exerting a negative pressure equal to the energy density of the electric field.
Conclusion:
Therefore, the force on one plate can be accounted for by thinking of the electrical field between the plates as exerting a negative pressure equal to the energy density of the electric field.
(b)
Answer to Problem 32.60AP
Explanation of Solution
Given info: The current density of capacitor plates is
The formula to calculate the force on one capacitor plate is,
Here,
Substitute
The formula to calculate the force per area acting on one sheet is,
Substitute
Conclusion:
Therefore, the force per area acting on one sheet due to the magnetic field is
(c)
Answer to Problem 32.60AP
Explanation of Solution
The formula to calculate the magnetic field due to positive sheet is,
The formula to calculate the magnetic field due to positive sheet is,
The formula to calculate the net magnetic field between the sheets is,
Substitute
The formula to calculate the net magnetic field outside the sheets is,
Substitute
Conclusion:
Therefore, the net magnetic field between the sheets is
(d)
Answer to Problem 32.60AP
Explanation of Solution
The formula to calculate energy density in the magnetic field between the sheets is,
Substitute
Conclusion:
Therefore, the energy density in the magnetic field between the sheets is
(e)
To show: The force on one sheet can be accounted for by thinking of the magnetic field between the sheets as exerting a positive pressure equal to its energy density.
Answer to Problem 32.60AP
Explanation of Solution
Given info: The current density of capacitor plates is
From part (b) the force per area acting on one sheet due to the magnetic field is
From part (d) the energy density in the magnetic field between the sheets is
Both the energy density in the magnetic field and the force per area acting on one sheet due to the magnetic field are equal. Hence, the force on one sheet can be accounted for by thinking of the magnetic field between the sheets as exerting a positive pressure equal to its energy density.
Conclusion:
Therefore, the force on one sheet can be accounted for by thinking of the magnetic field between the sheets as exerting a positive pressure equal to its energy density.
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Chapter 32 Solutions
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