Question

A long wire carrying a 5.0 A current perpendicular to the xy-plane intersects the x-axis at x= - 2.0 cm . A second, parallel wire carrying a 3.5 A current intersects the x-axis at x=+ 2.0cm .

A). At what point on the x-axis is the magnetic field zero if the two currents are in the same direction?

B). At what point on the x-axis is the magnetic field zero if the two currents are in opposite directions?

Answer 1

Answer:

a . 0.35cm

b.  11.33cm

Explanation:

a. Given both currents are in the same direction, the null point lies in between them. Let x be distance of N from first wire, then distance from 2nd wire is 4-x

#For the magnetic fields to be zero,the fields of both wires should be equal and opposite.They are only opposite in between the wires:

$\frac{\mu_oi_1}{2\pi x}=\frac{\mu_oi_2}{2\pi(4-x)}\\\\5/x=\frac{3.5}{4-x}\\\\x=2.35cm\\\\N=2.35-2=0.35cm$

Hence, for currents in same direction, the point is 0.35cm

b. Given both currents flow in opposite directions, the null point lies on the other side.

#For the magnetic fields to be zero,the fields of both wires should be equal and opposite.They are only opposite in outside the wires:

Let x be distance of N from first wire, then distance from 2nd wire is 4+x:

$\frac{\mu_oi_1}{2\pi(4+ x)}=\frac{\mu_oi_2}{2\pi x}\\\\5/(4+x)=\frac{3.5}{x}\\\\x=9.33cm\\\\N=9.33+2=11.33cm$

Hence, if currents are in opposite directions the point on x-axis is 11.33cm

Answer 2

(A) The currents are flowing in same direction and distance from zero magnetic field point is 0.35 cm.

(B) The currents are in opposite directions the point on x-axis is 11.33 cm.

Given data:

The magnitude of current in the wire is, I = 5.0 A.

The intersecting distance is, x' = -2.0 cm.

Magnitude of current in second wire is, I' = 3.5 A.

Intersecting distance from second wire is, x'' = +2.0 cm.

(A)

Since, both currents are in the same direction, the null point lies in between them. Let x be distance of N from first wire, then distance from 2nd wire is 4-x

So, For the magnetic fields to be zero, the fields of both wires should be equal and opposite. Then,

$\dfrac{\mu_{0} \times I}{2 \pi x}=\dfrac{\mu_{0} \times I'}{2 \pi (4 - x)}\\\\\\\dfrac{\mu_{0} \times I}{2 \pi x}=\dfrac{\mu_{0} \times I'}{2 \pi (4 - x)}\\\\\\\dfrac{I}{x}=\dfrac{I'}{(4 - x)}\\\\\\\dfrac{5}{x}=\dfrac{3.5}{(4 - x)}\\\\x =2.35 \;\rm cm$

So, point at which the magnetic field is zero is,

n = x - x'

n = 2.35 - 2.0

n = 0.35 cm

Thus, we can conclude that the currents are flowing in same direction and distance from zero magnetic field point is 0.35 cm.

(B)

Given both currents flow in opposite directions, the null point lies on the other side. Then the calculation is,

$\dfrac{\mu_{0} \times I}{2 \pi x}=\dfrac{\mu_{0} \times I'}{2 \pi (4 +x)}\\\\\\\dfrac{\mu_{0} \times I}{2 \pi x}=\dfrac{\mu_{0} \times I'}{2 \pi (4 +x)}\\\\\\\dfrac{I}{x}=\dfrac{I'}{(4+x)}\\\\\\\dfrac{5}{x}=\dfrac{3.5}{(4 + x)}\\\\x =9.33 \;\rm cm$

So, point at which the magnetic field is zero is,

n = x + x'

n = 9.33 + 2.0

n = 11.33 cm

Thus, we can conclude that the currents are in opposite directions the point on x-axis is 11.33 cm.

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