One long wire carries current 18.0 A to the left along the x axis. A second long

Question

One long wire carries current 18.0 A to the left along the x axis. A second long wire carries current 76.0 A to the right along the line (y = 0.280 m, z = 0). Where in the plane of the two wires is the total magnetic field equal to zero? A particle with a charge of -2.00 mu C is moving with a velocity of 1501 Mm/s along the line (y = 0.100 m, z = 0). Calculate the vector magnetic force acting on the particle. (Ignore relativistic effects.) A uniform electric field is applied to allow this particle to pass through this region undeflected. Calculate the required vector electric field.

Explanation / Answer

part a:

magnetic field due to both the wires will be along the same direction for points lying in between the two wires.

for all other points, magnetic fields will oppose each other.

as magnetic field due to an infinitely long wire is given by

B=mu*I/(2*pi*d)

where I=current through the wire

d=distance of the test point from the wire

the point where magnetic field due to two wires will be same in magnitude but opposite in direction should be closer to the wire carrying lesser current.

let the point is d m distance from wire along x axis.

then distance from wire along y=0.28 m is 0.28+d meters

hence equating both the fields:

mu*18/(2*pi*d)=mu*76/(2*pi*(d+0.28))

==>18*(d+0.28)=76*d

==>d=0.0869 m

part b:

magnetic field along the line y=0.1 m, z=0 :

distance from wire along x axis=0.1 m

distance from wire along 0.28 m =0.28-0.1=0.18 m

as the line lies between the two conductors, both the fields will be in same direction.

using right hand rule, direction of field will be along-ve z axis.
hence total field=(mu*18/(2*pi*0.1))+(mu*76/(2*pi*0.18))

=1.2044*10^(-4) T

field in vector notation: (-1.2044*10^(-4) k) T

then force=charge*(cross product between velocity vector and magnetic field vector)

=-2*10^(-6)*150*10^6*1.2044*10^(-4)*(cross product of i and -k)

=(-0.036132 j) N

part c:

force to be provided by the electric field so that net force will be zero=-(force by magnetic field)

=(0.036132 j) N

==>electric field=force/charge

=(-18066 j ) N/C

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