A long wire carrying a 5.0 A current perpendicular to the xy-plane intersects th

Question

A long wire carrying a 5.0 A current perpendicular to the xy-plane intersects the x-axis at x= -2.0cm. A second, parallel wire carrying a 2.5 A current intersects the x-axis at x=2.0cm. Part A At what point on the x-axis is the magnetic field zero if the two currents are in the same direction? Express your answer using two significant figures. Part B At what point on the x-axis is the magnetic field zero if the two currents are in opposite directions? Express your answer using two significant figures.

Explanation / Answer

Part A)

The magnetic field from a wire is found by the formula B = oI/2r

When the currents are in the same direction, the only location where the magnetic fields will cancel will be somewhere between the wires. If we call the distance from the 5 A wire 'x', then the distance from the 2.5 A wire is 4 - x

Then, the equations from the two wires can be set equal since they must cancel each other out.

oI1/2r1 = oI2/2r2

Since o, and 2 cancel, we get

I1/r1 = I2/r2

5/x = 2.5/(4-x)

Cross multiply

20 - 5x = 2.5x

x = 2.67 cm

Since x is the dstance between the wires from the 5 A location at -2 cm, then -2 + 2.67 = .67 cm

Thus, the location for zero B-field is at x = .67 cm

Part B)

When the currents flow in opposite direction, the only location where the B fields will cancel is beyond the 2 cm location since the field from the 5 A current will always be larger than the field from the 2.5 A current anywhere to the left of -2 cm location since it is larger inmagnitude. Also, between the wires, the fields will add and cannot cancel.

Thus, we can call the distance to the location of zero field x from the 2.5 A current and 4 + x from the 5 A current. The formula becomes

I1/r1 = I2/r2

5/(4 + x) = 2.5/x

Cross multiply

5x = 10 + 2.5x

x= 4 cm

Since x is the distance from the 2.5 A wire, the location on the x axis for cancellation is 2 + 4 = 6 cm

Thus, the location for zero B-field is at x = 6.0 cm

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