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A capacitor is constructed from two square, metallic plates of sides ℓ and separation d. Charges +Q and −Q are placed on the plates, and the power supply is then removed. A material of dielectric constant K is inserted a distance x into the capacitor as shown in Figure P20.85. Assume d is much smaller than x. (a) Find the equivalent capacitance of the device. (b) Calculate the energy stored in the capacitor. (c) Find the direction and magnitude of the force exerted by the plates on the dielectric. (d) Obtain a numerical value for the force when x = ℓ/2, assuming ℓ = 5.00 cm, d = 2.00 mm, the dielectric is glass (κ = 4.50), and the capacitor was charged to 2.00 × 103 V before the dielectric was inserted. Suggestion: The system can be considered as two capacitors connected in parallel.
Figure P20.85
(a)
The equivalent capacitance of the device.
Answer to Problem 85P
The equivalent capacitance of the device is
Explanation of Solution
Write the expression for the equivalent capacitance.
Here,
Write the expression for the area with dielectrics.
Here,
Write the equation for capacitance by using equation (II).
Here,
Write the expression for the area without dielectrics.
Write the equation for capacitance by using equation (IV).
Conclusion:
Substitute,
Thus, the equivalent capacitance of the device is
(b)
The energy stored in the capacitor.
Answer to Problem 85P
The energy stored in the capacitor is
Explanation of Solution
Write the expression for the stored energy.
Here,
Conclusion:
Substitute,
Thus, the energy stored in the capacitor is
(c)
The direction and the magnitude of the force exerted by the plates on the dielectrics.
Answer to Problem 85P
The direction and the magnitude of the force exerted by the plates on the dielectrics is
Explanation of Solution
Write the expression exerted force.
Conclusion:
Substitute,
For
For
The force is directed along positive x-direction.
Thus, the direction and the magnitude of the force exerted by the plates on the dielectrics is
(d)
The numerical value of force.
Answer to Problem 85P
The numerical value of force is
Explanation of Solution
Write the expression for the area.
Write the expression for the initial capacitance.
Here,
Write the expression for the charge.
Write the expression for the given condition.
Write the expression for the force by using (VI), (VII), (VIII), (IX) and (X).
Conclusion:
Substitute,
Thus, the numerical value of force is
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Chapter 20 Solutions
Principles of Physics: A Calculus-Based Text
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