At t = 0, the open switch in Figure P31.46 is thrown closed. We wish to find a symbolic expression for the current in the inductor for time t > 0. Let this current be called i and choose it to be downward in the inductor in Figure P31.46. Identify i1 as the current to the right through R1 and i2 as the current downward through R2. (a) Use Kirchhoff’s junction rule to find a relation among the three currents. (b) Use Kirchhoff’s loop rule around the left loop to find another relationship. (c) Use Kirchhoff’s loop rule around the outer loop to find a third relationship. (d) Eliminate i1 and i2 among the three equations to find an equation involving only the current i. (e) Compare the equation in part (d) with Equation 31.6 in the text. Use this comparison to rewrite Equation 31.7 in the text for the situation in this problem and show that
where R′ = R1R2/(R1 + R2).
Figure P31.46
(a)
The relation among the three currents by Kirchhoff’s junction rule.
Answer to Problem 46AP
The relation among the three currents by Kirchhoff’s junction rule are
Explanation of Solution
Given info: The figure that shows the given circuit is shown below.
Figure (I)
According to Kirchhoff’s junction rule, the total incoming currents are equal to the total outgoing currents at a junction.
From the circuit diagram equating the incoming currents to the outgoing current,
Here,
Conclusion:
Therefore, the relation among three currents by Kirchhoff’s junction rule are
(b)
The relationship between the given variables around the left loop by Kirchhoff’s loop rule.
Answer to Problem 46AP
The relationship between the given variables around the left loop by Kirchhoff’s loop rule is
Explanation of Solution
Given info: The figure that shows the given circuit is shown in figure (I).
According to Kirchhoff’s loop rule, the sum of all the voltage across all the elements in a loop must be zero.
From the circuit diagram equating the voltage across the elements in the left loop is equal to zero.
Here,
Conclusion:
Therefore, the relationship between the given variables around the left loop by Kirchhoff’s loop rule is
(c)
The relationship between the given variables around the outer loop by Kirchhoff’s loop rule.
Answer to Problem 46AP
The relationship between the given variables around the outer loop by Kirchhoff’s loop rule is
Explanation of Solution
Given info: The figure that shows the given circuit is shown in figure (I).
According to Kirchhoff’s loop rule, the sum of all the voltage across all the elements in a loop must be zero.
From the circuit diagram equating the voltage across the elements in the outer loop is equal to zero.
Conclusion:
Therefore, the relationship between the given variables around the outer loop by Kirchhoff’s loop rule is
(d)
The equation that involve only current
Answer to Problem 46AP
The equation that involve only current
Explanation of Solution
Given info: The figure that shows the given circuit is shown in figure (I).
From equation (1), the expression for the
Substitute
From equation (2), the expression for the
Substitute
Equate equation (3) and equation (4) for
Further solve the above equation,
Assume
Substitute
Thus, the require equation in term of current
Conclusion:
Therefore, the equation that involve only current
(e)
The comparison of equation for part (d) with the equation
Answer to Problem 46AP
The equation
Explanation of Solution
From the textbook the equation
From the part (d), the equation is given as,
Since both the equation shown above are same therefore their solution are also same.
The solution of the equation
Similarly rewrite the equation
Substitute
Conclusion:
Therefore, the equation
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Chapter 31 Solutions
Physics for Scientists and Engineers with Modern Physics
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