# Surface Charge Density on an infinite sheet

A small, nonconducting ball of mass 1.4E-6 kg and charge 1.9E-8 C hangs from an insulating thread that makes an angle of 32 degrees with a vertical, uniformly charged nonconducting sheet. Considering the gravitational force on the ball and assuming the sheet extends far vertically and into and out of the page, calculate the surface charge density of the sheet?

So that is the question.

Basically, there is an infinitely long sheet and at some point is a hanging ball of positive charge given that hangs 32 degrees to the right of the sheet.

I'm not sure at all where to start on this question, and do not know which equations to use. Any help would be greatly appreciated. I simply know that there is a gravitational force down on the ball and an electric force pointing right on the ball, due to the electric field.

## The Attempt at a Solution

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A small, nonconducting ball of mass 1.4E-6 kg and charge 1.9E-8 C hangs from an insulating thread that makes an angle of 32 degrees with a vertical, uniformly charged nonconducting sheet. Considering the gravitational force on the ball and assuming the sheet extends far vertically and into and out of the page, calculate the surface charge density of the sheet?

So that is the question.

Basically, there is an infinitely long sheet and at some point is a hanging ball of positive charge given that hangs 32 degrees to the right of the sheet.

I'm not sure at all where to start on this question, and do not know which equations to use. Any help would be greatly appreciated. I simply know that there is a gravitational force down on the ball and an electric force pointing right on the ball, due to the electric field.

## The Attempt at a Solution

First off, do you know the equation for the electric field for an infinitely long sheet of charge? If not, you need to calculate that solution probably using Gauss' law. Next, recognize that electric fields generate a force on charge and gravitational fields generate a force on mass. The pith ball has both mass and charge. You need to find what electric field (which is a function of charge density for an infinite sheet) would produce enough force to allow the ball to levitate at the given angle. You'll need to fall back onto your mechanics problems with rope tension and so on.

The equation for the electric field for an infinitely long sheet of charge is simply
E = (surface charge density)/(2 * epsilon not).

I do not understand how the calculation of gravity plays a role, since the sheet would produce a field horizontally, which would push the pith ball out to an angle of 32 degrees. Gravity points downwards, and has no effect on the pith ball. Why would gravity matter?

The equation for the electric field for an infinitely long sheet of charge is simply
E = (surface charge density)/(2 * epsilon not).

I do not understand how the calculation of gravity plays a role, since the sheet would produce a field horizontally, which would push the pith ball out to an angle of 32 degrees. Gravity points downwards, and has no effect on the pith ball. Why would gravity matter?
The tension in the string holding the pith ball has an x-component from the electric field and a y-component from gravity. Thus, you can use the angle and gravity to find the total tension, and then use the total tension to find the x-component(electric force).

I understand that part, thank you!

So I calculated the Force due to gravity and got
Fg = 1.3734E-5 N

Using that and the angle, I got the Fnet (total tension)
Fnet = 1.6195E-15 N

Knowing Fnet, I was able to find the Force due to the electric field in the x direction (let's call it Fe for Force due to electric field):
Fe = 8.582E-16 N

Using my known force due to the electric field, how can I compute the surface charge density? I do know there is an equation E = F/q.

I understand that part, thank you!

So I calculated the Force due to gravity and got
Fg = 1.3734E-5 N

Using that and the angle, I got the Fnet (total tension)
Fnet = 1.6195E-15 N

Knowing Fnet, I was able to find the Force due to the electric field in the x direction (let's call it Fe for Force due to electric field):
Fe = 8.582E-16 N

Using my known force due to the electric field, how can I compute the surface charge density? I do know there is an equation E = F/q.
Use that equation except substitute in the equation for the electric field of an infinite sheet of charge(which has charge density in it). Next, solve for charge density.