Consider an infinitely long, very wide planar sheet of parallel wires lying in t

Question

Consider an infinitely long, very wide planar sheet of parallel wires lying in the x - y plane, with each wire carrying a current I in the + y direction. The sheet has N wires and a total width L . Consider the magnetic field created by all this current at an arbitrary point P a distance d from the sheet, where and P is very far from the edges.

(Due to the sheet’s symmetry, the magnetic field values at P and P’, equal distances above and below the sheet, must have equal magnitudes and must have opposite x and z components. (Imagine rotating the entire sheet 180 around its middle, along a line going into the page. The sheet would look unchanged, so the rotated magnetic field vectors must look unchanged, too.)

a)Use Gauss’ law for magnetism to show that the x-component (perpendicular to the sheet) of the magnetic field at P must be zero. For your “Gaussian surface,” use a rectangular box whose cross-section is the dotted line in the figure.

b)Use Ampere’s law to show that the y -component (into the page, the direction of the current flow) of the magnetic field at P must be zero. For your “Amperian loop,” use a rectangular loop that goes into the page through P, turns 90 and goes straight through the sheet for adistance of 2d , turns 90 again and comes back out of the page through P ’, and then turns once more and goes back through the sheet to where it started. (On the diagram above, the “end view” would look like a straight line from P to P’.)

c)Use Ampere’s law to find the x-component (parallel to the sheet, perpendicular to the current) of the magnetic field at P in terms of I , N , L , and d . For your “Amperian loop,” use the dashed rectangle shown.

d) Put together your results from parts (a-c) to find the magnetic field vector of the sheet of current. How does it depend on distance from the sheet? How does your result compare to the electric field of a large flat sheet of charge?

Explanation / Answer

given, infinitely long very wide sheet of parallel wires in x-y plane with current I in each wire in +y direction

N wires and width = L

consider point P

a. for the given rectangular box, let its length be l, depth (into the page be d)

then, the surface of the box parallel to the page will have no magnetic field compoentn perpendicular to them because the current is along that direction

hence

the magnetic field can exist in x and z direction

so for a constant megnetic field ( as this box is far from the edges)

form gauss' law

2B.(ld)k = 0

k is unit vecotr in z direction

let B = Bxi + Bzk

then

(Bxi + Bzk).(ld)k = 0

for this dot product to be zero

Bx = 0

hence there is no x component of magnetic field

b. the y magnetic field has to be zero because curernt is flowing in that direction

but lets assume there exists a y directed magnetic field

then consideringf the amperean loop such that it lies in xy plane

then

By*L = mu*i ( L is length of the loop)

now, curent flowing through the loop = 0 ( in xy plane, as all current flows across xz plane)

hence

By = 0

c. considering Bz only now

form amperes law ( using loop given in the figure)

2Bz*l = Nl*i*mu/L ( where mu is permeability of free space)

hence Bz = Ni*mu/2L

d. magnetic field at any point, B = Ni*mu k /2L for z > 0 and Ni*mu (-k)/2L for z < 0

this magnetic field is constant and does not depend on distance from the sheet

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