(a)
The relation between the three currents
(a)
Answer to Problem 60P
The three currents are related by the equation
Explanation of Solution
Figure.1 shows the flow of currents in the circuit.
From figure.1,
The current through the resistor
Conclusion:
Therefore, the equation relating the three currents in the circuit is
(b)
The relation using the loop in the left hand side
(b)
Answer to Problem 60P
The relation obtained from the left hand side loop is
Explanation of Solution
According to Kirchhoff’s voltage rule, the algebraic sum of all the voltages in any closed loop in a circuit is zero.
Consider the direction of current from positive terminal of battery to the negative terminal to write the Kirchhoff’s voltage rule using figure.1.
The equation for the voltage across the resistor
Conclusion:
Write the algebraic sum of each of the voltages to obtain the required equation.
Therefore, the relation obtained from the left hand side loop is
(c)
The relation using the outer loop
(c)
Answer to Problem 60P
The relation obtained from the outer loop is
Explanation of Solution
According to Kirchhoff’s voltage rule, the algebraic sum of all the voltages in any closed loop in a circuit is zero.
Consider the direction of current from positive terminal of battery to the negative terminal to write the Kirchhoff’s voltage rule using figure.1.
The equation for the voltage across the resistor
Conclusion:
Write the algebraic sum of each of the voltages to obtain the required equation.
Therefore, the relation obtained from the outer loop is
(d)
An equation having only
(d)
Answer to Problem 60P
The relation having only
Explanation of Solution
Substitute equation (I) in equation (II).
Substitute equation (I) in equation (III).
Compare equation (III) and equation (IV).
Conclusion:
Substitute
Therefore, the relation having only
(e)
The equation of current with respect to time
(e)
Answer to Problem 60P
The equation of current with respect to time is
Explanation of Solution
The Kirchhoff’s loop rule for the reference equation is
Write the equation for the solution of equation (VI).
Here,
Conclusion:
Substitute equation (VIII) in equation (VII0.
Here,
Therefore, equation of current with respect to time is
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Chapter 23 Solutions
Principles of Physics: A Calculus-Based Text
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