Concept explainers
(a)
The rate of deliver of energy
(a)
Answer to Problem 50P
The rate of deliver of energy is
Explanation of Solution
The rate of deliver of energy is the power of the battery. Write the equation for the power of the battery.
Here,
Conclusion:
Substitute
Therefore, the rate of deliver of energy by the battery is
(b)
The power delivered to the resistance of the coil
(b)
Answer to Problem 50P
The power delivered to the resistance of the coil is
Explanation of Solution
Write the equation for the power delivered to the resistance of the coil.
Here,
Write the equation for the voltage across the resistance.
Here,
Conclusion:
Substitute
Therefore, the power delivered to the resistance of the coil is
(c)
The rate of energy storage
(c)
Answer to Problem 50P
The rate of storage of energy is
Explanation of Solution
Consider the inductor being ideal and connect in series with an ideal resistor. According to Kirchhoff’s voltage rule, the algebraic sum of all the voltages in any closed loop in a circuit is zero.
Write the equation for the algebraic sum of the voltages across the coil.
Here,
The rate of storage of energy is the power. Write the equation for the power stored in the inductor.
Here,
Conclusion:
Rearrange equation (VI) and solve for
Substitute
Therefore, the rate of storage of energy in the magnetic field is
(d)
The relation between the three power values
(d)
Answer to Problem 50P
The power from the battery is the sum of the power across the internal resistance and the power in the magnetic field.
Explanation of Solution
From equation (II), the battery is delivering energy at a rate of
From equation (V), the power delivered to the resistance of the coil is
From equation (VIII), the rate of storage of energy in the magnetic field is
From the value of different powers given in equation (II), equation (V) and equation (VIII), it can be inferred that
Conclusion:
Therefore, the power delivered from the battery is the sum of the power delivered to the internal resistance and the power stored in the magnetic field.
(e)
The validity of the relation
(e)
Answer to Problem 50P
Yes, it is valid in other instants as well
Explanation of Solution
The relation between the powers is that the power from the battery is the sum of the power across the internal resistance and the power in the magnetic field.
At any instant, the power generated by the battery is the sum of the power delivered to the internal resistance and the power stored in the magnetic field.
Conclusion:
Therefore, it is true that the relation between the power is valid at any istant.
(f)
The relation between the power at given instants
(f)
Answer to Problem 50P
The power delivered to the resistance is zero at
Explanation of Solution
From equation (III) and equation (IV), write the equation for the power delivered to the resistance.
Here,
Write the equation for the power delivered by the magnetic field.
Here,
Conclusion:
Immediately after
After some time, the current does not change anymore and hence there is no power being stored in the magnetic field. All the power from the battery is delivered to the resistance of the coil.
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Chapter 23 Solutions
Principles of Physics: A Calculus-Based Text
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