Physics for Scientists and Engineers
Physics for Scientists and Engineers
6th Edition
ISBN: 9781429281843
Author: Tipler
Publisher: MAC HIGHER
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Chapter 29, Problem 27P

(a)

To determine

To Find: The peak current in the circuit using Kirchhoff’s loop rule and trigonometric identity.

(a)

Expert Solution
Check Mark

Answer to Problem 27P

  ipeak=0.3464A

Explanation of Solution

Given:

Potential difference across terminals of one of the generators, V1=(5 V)cos(ωtα)

Potential difference across terminals of one of the other generators, V2=(5 V)cos(ωt+α)

Where, α=π6

Resistance, R=25Ω

Formula used:

Kirchhoff’s Voltage law:

Sum of the potential in a closed loop circuit is zero.

Calculation:

  Physics for Scientists and Engineers, Chapter 29, Problem 27P , additional homework tip  1

ApplyingKirchhoff’s voltage law:

  25i=5cos(ωtα)+5(ωt+α)25i=(cos(ωtα)+(ωt+α))5i=2cosωtcosαi=25cosωtcosπ6i=35cosωtipeak=35Aipeak=0.3464A

Conclusion:

  ipeak=0.3464A is the peak current in the circuit using Kirchhoff’s loop rule and trigonometric identity

(b)

To determine

To Find: The peak current in the circuit by using a phasor diagram.

(b)

Expert Solution
Check Mark

Answer to Problem 27P

  ipeak=0.3464A

Explanation of Solution

Given:

Potential difference across terminals of one of the generators, V1=(5 V)cos(ωtα)

Potential difference across terminals of one of the other generators, V2=(5 V)cos(ωt+α)

Where, α=π6

Resistance, R=25Ω

Formula used:

Vector formula

  |V|=|V1|2+|V2|2+2|V1||V2|cos2α

Where,

  |V| is the resultant vector

  V1 and V2 are the two vectors and 2α is the angle between them.

Calculation:

  Physics for Scientists and Engineers, Chapter 29, Problem 27P , additional homework tip  2

  θ1 and θ2 can be given as:

  θ1=ωtαθ2=ωt+αθ2θ2=ωt+αωt+αθ2θ2=2α

Angle between the vectors V1 and V2 is 2α

Magnitude of V can be calculated as:

  |V|=|V1|2+|V2|2+2|V1||V2|cos2α|V|=52+52+2×5×5cos2α|V|=52(1+cos2α)|V|=52×2cos2α|V|=10cosα

But,

  |V|=IpeakRIpeakR=10cosαIpeak=10cosα25Ipeak=2cosα5

It is given that

  α=π6

Therefore,

  Ipeak=25cos(π6)Ipeak=25×32Ipeak=0.3464A

Conclusion:

  ipeak=0.3464A is the peak current in the circuit by using a phasor diagram.

(c)

To determine

To Find: The current in the resistor.

(c)

Expert Solution
Check Mark

Answer to Problem 27P

  250[12cosωtsinωt]

Explanation of Solution

Given:

Potential difference across terminals of one of the generators, V1=(5 V)cos(ωtα)

Potential difference across terminals of one of the other generators, V2=(7 V)cos(ωt+α)

Where, α=π4

Resistance, R=25Ω

Formula used:

Kirchhoff’s loop rule

Sum of the potential in the circuit is zero.

Calculation:

ApplyingKirchhoff’s loop rule

  iR=V1+V2

  iR=5cos(ωtα)+7cos(ωt+α)iR=5cosωtcosα+5sinωtsinα+7cosωtcosα7sinωtsinαiR=12cosωtcosα2sinωtsinα

For,

  α=π4cosπ4=sinπ4=12

Therefore,

  i×25=12[12cosωtsinωt]i=250[12cosωtsinωt]

Conclusion:

The current in the resistor is 250[12cosωtsinωt] .

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