# Physics: Find the electric field on any point on a y-axis and find the electric field as a function of the radial distance.

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5.) A square of paper measuring a on a side carries a total charge +q which is uniformly distributed over its surface. The square lies in the x-y plane with its center at the origin and its sides parallel to the coordinate axes. Find the electric field on any point on the y-axis with y > a. Show that this square looks like a point charge for the case y>> a.

9.) The electric charge of the proton is not concentrated at a point but rather distributed over a volume. According to experimental investigations at the Stanford Linear Accelerator, the charge distribution of the pro- ton can be approximated by an exponential function

e

p= ____ e^((-r)/b)

8(pi)b

where r is the radial position inside the proton and b is a constant equal to 0.23 x 10^(-15)m. Find the electric field as a function of the radial distance. What is the magnitude of the electric field at r =1 x 10^(-15)m? Compare the electric field strength you find to that of a point charge of magnitude e. At what distances r do these two differ by 10% or more?

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The solution finds the electric field on any point on a y-axis.

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We begin by finding the field of a infinitely thin strip of charge (with linear charge density), of length 2L, parallel to the x axis at distance y off center.

Each such charge equals

(1.1)

Each of these charges contributes an electric field of:

(1.2)

Where and k is Coulomb constant

Due to symmetry, the horizontal contributions at point P from all these small charges will cancel itself and we are left with the vertical component:

(1.3)

Now we have to integrate all these contributions from to , but due to the parity of the integrand we need to integrate between to and multiply by 2:

(1.4)

To solve this integral we use the substitution:

(1.5)

Then:

Putting this together:

(1.6)

Note that

(1.7)

Hence the integral becomes trivial:

(1.8)

And its solution is:

(1.9)

If we use the identity (see appendix, equation 1.38)

(1.10)

We get:

(1.11)

This is the electric field of a single thin strip of charge of length 2L at distance y off its center.

Now we want to find the electric field along the y-axis, of a square plate of side length of 2L, centered about the origin, parallel ...

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