Question

Three infinite straight wires are fixed in place and aligned parallel to the z-axis as shown. The wire at (x,y) = (-13.5 cm, 0) carries current I₁ = 3.2 A in the negative z-direction. The wire at (x,y) = (13.5 cm, 0) carries current I₂ = 1.4 A in the positive z-direction. The wire at (x,y) = (0, 23.4 cm) carries current I₃ = 5.1 A in the positive z-direction.

1)

What is B_x(0,0), the x-component of the magnetic field produced by these three wires at the origin?T

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2)

What is B_y(0,0), the y-component of the magnetic field produced by these three wires at the origin?T

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3)

What is F_x(1), the x-component of the force exerted on a one meter length of the wire carrying current I₁?N

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4)

What is F_y(1), the y-component of the force exerted on a one meter length of the wire carrying current I₁?N

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5)

What is F_x(2), the x-component of the force exerted on a one meter length of the wire carrying current I₂?

Answer 1

Please refer to this image first and follow the solution below.

1) According to the right hand thumb rule, the x contribution of magneteic field is from I₃ wire.

So,

The values of

u₀=1.25663706 × 10^-6  m kg s^-2A^-2

I₃= 5.1 A, a₃=0.234 m

Putting the values, we get B_x(0,0)= 4.37 X 10^-6 T

2) I₁ and I₂ contribute to the y component of the magnetic field.

So,

a₁=a₂=0.135 m

I₁= 3.2 A and I₂= 1.4 A

Putiing the values, we get B_y(0,0)=-6.84 X 10 ^-6 T

The negative value indicates the negative y direction(by right hand thumb rule)

3) To find the x component of Force, first we should calculate the y component of magnetic field on wire I₁ due to wire

I₂ and I₃ (Force and magnetic field are perpendicular to each other)

so,

d₁₂= 13.5+13.5=27 cm

Now,

and

Put the values and get the answer.

4) Similarly for y component of force, we need to calculte x component of magnetic field.

The x component of magnetic filed on wire I₁ is due to I₃ only.

So,

And

We have all the values, put them in the equation above and get the answer.

5) First we need to find the y component of magnetic filed on wire I₂ due to I₁ and I₃

d₂₁= 13.5+13.5=27 cm

Now,

and

Put the values and get the answer.