Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
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Chapter 8, Problem 10P

(a)

To determine

Calculate the force per unit length on L2 due to L1. Also, state whether the force is attractive or repulsive.

(a)

Expert Solution
Check Mark

Answer to Problem 10P

The force per unit length on L2 due to L1 is F21=1axmN/m and the force is repulsive.

Explanation of Solution

Calculation:

Write the general expression to calculate the force.

F=IL×B

or

FL=I1al×B2=μoI1I2al×aϕ2πρ{B2=μoI22πρaϕ}        (1)

Here,

μo is the permeability of free space,

I1 and I2 are the currents, and

B2 is the magnetic flux density.

To find the distance ρ at points (0, 4) and (0, 0).

ρ=(00)2+(04)2=4

Using equation (1), the force per unit length on L2 due to L1 is calculated as follows.

F21=(4π×107)(100)(200)az×(ay)2π(4){ρ=4,μo=4π×107}=1×103ax{az×(ay)=ax}F21=1axmN/m{1m=103}

The force per unit length on L2 due to L1 is repulsive because the current 200 A and –100 A flows in opposite direction.

Conclusion:

Thus, the force per unit length on L2 due to L1 is F21=1axmN/m and the force is repulsive.

(b)

To determine

Calculate the force per unit length on L1 due to L2. Also, state whether the force is attractive or repulsive.

(b)

Expert Solution
Check Mark

Answer to Problem 10P

The force per unit length on L1 due to L2 is F12=1axmN/m and the force is repulsive.

Explanation of Solution

Calculation:

Refer to Part (a),

The force per unit length on L2 due to L1 is F21=1axmN/m.

The force per unit length on L1 due to L2 is calculated as follows.

F12=F21

Substitute 1axmN/m for F21 in above equation.

F12=1axmN/m

The force per unit length on L1 due to L2 is repulsive because the current 200 A and –100 A flows in opposite direction.

Conclusion:

Thus, the force per unit length on L1 due to L2 is F12=1axmN/m and the force is repulsive.

(c)

To determine

Calculate the force per unit length on L3 due to L1. Also, state whether the force is attractive or repulsive.

(c)

Expert Solution
Check Mark

Answer to Problem 10P

The force per unit length on L3 due to L1 is F31=(0.96ax+0.72ay)mN/m and the force is repulsive.

Explanation of Solution

Calculation:

The unit vector aϕ is determined as follows,

aϕ=(0,0)(3,4)(03)2+(04)2=3ax4ay9+16=3ax4ay5=3ax54ay5

In equation (1), al×aϕ is determined as follows.

al×aϕ=az×(3ax54ay5)=35ay+45ax

To find the distance ρ at points (3, 4) and (0, 0).

ρ=(03)2+(04)2=9+16=25=5

Using equation (1), the force per unit length on L3 due to L1 is calculated as follows.

F31=(4π×107)(100)(300)(35ay+45ax)2π(5){al×aϕ=35ay+45ax,ρ=5,μo=4π×107}=1.2×103(45ax35ay)F31=(0.96ax+0.72ay)mN/m{1m=103}

The force per unit length on L3 due to L1 is repulsive because the current 300 A and –100 A flows in opposite direction.

Conclusion:

Thus, the force per unit length on L3 due to L1 is F31=(0.96ax+0.72ay)mN/m and the force is repulsive.

(d)

To determine

Calculate the force per unit length on L3 due to L1 and L2. Also, state whether the force is attractive or repulsive.

(d)

Expert Solution
Check Mark

Answer to Problem 10P

The force per unit length on L3 due to L1 and L2 is F3=4.96ax+0.72aymN/m. The force per unit length on L3 due to L1 is repulsive and the force per unit length on L3 due to L2 is attractive.

Explanation of Solution

Calculation:

To find the force per unit length on L3 due to L2:

To find the distance ρ at points (3, 4) and (0, 4).

ρ=(03)2+(44)2=9+0=3

Using equation (1), the force per unit length on L3 due to L2 is calculated as follows.

F32=(4π×107)(200)(300)az×ay2π(3){ρ=3,μo=4π×107}=4×103ax{az×ay=ax}F32=4axmN/m{1m=103}

The force per unit length on L3 due to L1 and L2 is,

F3=F31+F32

Substitute (0.96ax+0.72ay)mN/m for F31 and 4axmN/m for F32 in above equation.

F3=(0.96ax+0.72ay)mN/m4axmN/m=4.96ax+0.72aymN/m

The force per unit length on L3 due to L1 is repulsive because the current 300 A and –100 A flows in opposite direction. The force per unit length on L3 due to L2 is attractive because the current 300 A and 200 A flows in same direction

Conclusion:

Thus, the force per unit length on L3 due to L1 and L2 is F3=4.96ax+0.72aymN/m. The force per unit length on L3 due to L1 is repulsive and the force per unit length on L3 due to L2 is attractive.

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Chapter 8 Solutions

Elements Of Electromagnetics

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