Two very long current-carrying wires are a distance d apart, as shown. I2 = 4I1

Question

Two very long current-carrying wires are a distance d apart, as shown. I2 = 4I1 Where is the magnetic field equal to zero? Show all work. Suppose the current direction of 11 is opposite the original direction. Where is the magnetic field equal to zero? Show work. For the two currents in opposite direction, what is the force of top wire on the bottom wire? What is the force of the bottom wire on the top wire? Show work and explain. For the two currents in the same direction, what is the force of top wire on the bottom wire? What is the force of the bottom wire on the top wire? Show work and explain.

Explanation / Answer

a. magnetic field formula B = uoi/2pi R

force between two parallel wires is given by F = u0i1i2L/2pid

where u0 = constnat = 4pi*10^-7 H/m

i1, i2 are the currents

L s the length o the wire

d is the distqance between these two wires

if current are in same directiion, there exists , they exert attractive force on each other

and if currents are in opposite direction, they exert repulsive force

so now by

part A: let field B e zero at x units from I1 then it is at (x+d) from I2

so at this Position B1 = B2

uoI1/2pix = uoI2/2pi (d-x)

I1/x = 4 I1/(d-x)

d-x/x = 4

d-x = 4 x

5x = d

x = d/5 at a location d/5 from top wire Field is zero

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at same distance from bottom wire

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F = 4*3.14 e-7 * I1 * 4I1 /(2*3.14 * d)

Ftop = 4pi e -7 * 4I^2/2pi d attractvire force

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Fbotttom = - 4pi e -7 * 4I^2/2pi d

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F = 4*3.14 e-7 * I1 * 4I1 /(2*3.14 * d)

Ftop = 4pi e -7 * 4I^2/2pi d repulsive e force

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Fbotttom = - 4pi e -7 * 4I^2/2pi d

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