Hello, I need help with Part C of this physics problem. Part A gives all the inf

Question

Hello,

I need help with Part C of this physics problem. Part A gives all the information and Part B (not pictured) was just a writing prompt. Don't mind the two lines on the graph as those are my wrong attempts.

Thank you!

An infinitely long sheet of charge of width L lies in the xy-plane between x = -L/2 and x = L/2. The surface charge density is zeta. Derive an expression for the electric field at height z above the centerline of the sheet. (Assume that z 0.) Express your answer in terms of the variables zeta, L, z, unit vector , and appropriate constants. Draw a graph of field strength E versus z.

Explanation / Answer

Let us find the electric field at a point P which is at a distance z from the center line.

Divide the given charged sheet into infinite sheets parallel to the y-axis each having width dx.

Consider one such sheet at a distance x from the y-axis.
Since the width of this sheet is an infinitesimally small quantity dx, the electric field due to this sheet at P is the same as due to a line charge.

Now find the linear charge density of this line charge as follows :

Consider a rectangle of dimensions L x 1m from the original sheet such that L is along the x-axis and 1m is along the Y-axis.
Area of this rectangle = Lx1 = L
So charge on this rectangle = L*eta

When this sheet is divided into infinite line charges, this rectangle also gets divided into infinite line charges, each of length 1m.
Charge contained in this 1m length = L*eta*(dx / L) = eta*dx

So linear charge density of the line charge = eta*dx.

Now field due to the line charge at P,
dE = (eta*dx) / [ 2*pi*e*sqrt(x^2 + h^2) ]

By symmetry, we can see that the horizontal component of this field will get canceled by the
mirror element of the this element.

So only the vertical component contributes to the total field at P.

Vertical component = (eta*dx) / [ 2*pi*e*sqrt(x^2 + z^2) ] * z / sqrt(x^2 + z^2)
(e = epsilon)
Integrate this expression from x = -L/2 to x = L/2 to get the net field at P.

Field at P = E(z) = (eta / pi*e) * tan^-1 (L/2z)

(b)

For z<<L,
E(z) = (eta / pi*e) * tan^-1 (infinity)
= (eta / pi*e) * pi*2
= eta / 2*e .....As expected, this is the field due to an infinite sheet

For z>>L,
E(z) = eta*L / 2z ....As expected, this is the field due to an infinite wire
(To get this result, use tan^-1 x approaches x as x approaches zero)

Hope this helps.

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