Physics for Scientists and Engineers with Modern Physics
Physics for Scientists and Engineers with Modern Physics
10th Edition
ISBN: 9781337553292
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 31, Problem 46AP

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

i ( t ) = ε R 1 [ 1 e ( R / L ) t ]

where R′ = R1R2/(R1 + R2).

Figure P31.46

Chapter 31, Problem 46AP, At t = 0, the open switch in Figure P31.46 is thrown closed. We wish to find a symbolic expression

(a)

Expert Solution
Check Mark
To determine

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 i1=i2+i .

Explanation of Solution

Given info: The figure that shows the given circuit is shown below.

Physics for Scientists and Engineers with Modern Physics, Chapter 31, Problem 46AP

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,

i1=i2+i

Here,

i1 is the current flow through the resistance R1 .

i2 is the current flow through the resistance R2 .

i is the current flow through the inductor.

Conclusion:

Therefore, the relation among three currents by Kirchhoff’s junction rule are i1=i2+i .

(b)

Expert Solution
Check Mark
To determine

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 εi1R1i2R2=0 .

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.

εi1R1i2R2=0 (1)

Here,

ε is the emf of the battery.

Conclusion:

Therefore, the relationship between the given variables around the left loop by Kirchhoff’s loop rule is εi1R1i2R2=0 .

(c)

Expert Solution
Check Mark
To determine

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 εi1R1Ldidt=0 .

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.

εi1R1Ldidt=0 (2)

Conclusion:

Therefore, the relationship between the given variables around the outer loop by Kirchhoff’s loop rule is εi1R1Ldidt=0 .

(d)

Expert Solution
Check Mark
To determine

The equation that involve only current i

Answer to Problem 46AP

The equation that involve only current i is ε'iR'Ldidt=0 .

Explanation of Solution

Given info: The figure that shows the given circuit is shown in figure (I).

From equation (1), the expression for the

εi1R1i2R2=0

Substitute i2+i for i1 in above equation and rearrange for i2 .

ε(i2+i)R1i2R2=0i2R1+iR1+i2R2=εi2(R1+R2)=εiR1i2=εiR1R1+R2 (3)

From equation (2), the expression for the

εi1R1Ldidt=0

Substitute i2+i for i1 in above equation and rearrange for i2 .

ε(i2+i)R1Ldidt=0(i2+i)R1+Ldidt=ε(i2+i)=εLdidtR1i2=εLdidtR1i (4)

Equate equation (3) and equation (4) for i2 .

εLdidtR1i=εiR1R1+R2(εiR1R1+R2+i)R1=εLdidt(εiR1+(R1+R2)iR1+R2)R1=εLdidt(ε+iR2R1+R2)R1=εLdidt

Further solve the above equation,

ε(ε+iR2R1+R2)R1=Ldidtε(R1+R2)(ε+iR2)R1R1+R2=LdidtεR2iR1R2R1+R2=Ldidtε(R2R1+R2)i(R1R2R1+R2)Ldidt=0

Assume ε'=ε(R2R1+R2) and R'=(R1R2R1+R2) .

Substitute ε' for ε(R2R1+R2) and R' for (R1R2R1+R2) in equation ().

ε'iR'Ldidt=0

Thus, the require equation in term of current i is ε'iR'Ldidt=0 .

Conclusion:

Therefore, the equation that involve only current i is ε'iR'Ldidt=0 .

(e)

Expert Solution
Check Mark
To determine

The comparison of equation for part (d) with the equation 31.6 in the text use this comparison rewrite the equation 31.7 in the text for the text and show that i(t)=εR1[1e(R'/L)t]

Answer to Problem 46AP

The equation 31.6 in the text and the equation in part (d) are similar also the equation 31.7 can be rewrite for this situation is i(t)=ε'R'[1e(R'/L)t] and it showed that i(t)=εR1[1e(R'/L)t] .

Explanation of Solution

From the textbook the equation 31.6 is given as,

εiRLdidt=0

From the part (d), the equation is given as,

ε'iR'Ldidt=0

Since both the equation shown above are same therefore their solution are also same.

The solution of the equation 31.6 is given in the text as equation 31.7 that is,

i(t)=εR1[1e(R'/L)t]

Similarly rewrite the equation 31.7 for the part (d) equation.

i(t)=ε'R'[1e(R'/L)t]

Substitute ε' for ε(R2R1+R2) and R' for (R1R2R1+R2) in equation ().

i(t)=(ε(R2R1+R2))((R1R2R1+R2))[1e(R'/L)t]i(t)=εR1[1e(R'/L)t]

Conclusion:

Therefore, the equation 31.6 in the text and the equation in part (d) are similar also the equation 31.7 can be rewrite for this situation is i(t)=ε'R'[1e(R'/L)t] and it showed that i(t)=εR1[1e(R'/L)t] .

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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 i, as the current to the right through R, and iz as the current downward through R. (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 i, and i, 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 R Figure P31.46 to rewrite Equation 31.7 in the text for the situation in this problem and show that i(t) R, where R' = R,R,/(R, + R,).
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Chapter 31 Solutions

Physics for Scientists and Engineers with Modern Physics

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