Electrical Fields

Important

Electric fields are created by charge

Electric Field Strength

Equation

where:

is the electric field ( volts per meter, or newtons per coulomb)
is Coulomb's constant ()
is the charge ()
is the distance between the two charges ()

Remark

is an "imaginary" charge (test charge) to measure what the strength WOULD be with that charge.

Example

A Van der Graaf generator is charged to $µ$ . What is the electric field away?

solution

µ

Recall that a negative value would mean the vector points away.

Electrical Force (Coulomb's Law)

Equation

and we tag on the unit vector at the end to give the direction, but usually this shouldn't be done explicitly.

Remark

Sometimes we write as , where is the constant of permittivity of free space, which is , which gives

so sometimes Coulomb's Law is expressed as

Remark

Sometimes, it's good to explicitly denote the directions of force, for example,

which represents the force object 1 exerts on object 2.

Example

An electron is from a proton. What is the force between them?

solution Recall from the Cupboard Formulae:

Electron charge:
Proton charge:

Note that a positive value would mean repulsion.

Force on a Moving Charge

Recall from Cupboard Formulae that , and . Since m

and we can figure out lots of stuff with this.

Charge Distributions

Equation

When a uniform charge is uniformly distributed along a thin rod of length , the linear charge density is:

similarly, for a surface with area :

and for volume , which is analogous to mass density:

Finding the Electrical Force on a Point Charge From a Distribution

Abstract

Find the linear density
Divide that continuous distribution into differential elements, :
integrate the differential force over all the differential charge elements of the objects.

Example - Force From a Line of Charge

A total positive charge of $µ$ is evenly distributed on a straight thin rod of length . A positive point charge, , is located a distance of above the midpoint of the rod. What is the electrical force on the point charge?

solution

Assume the rod is parallel to the x-axis and its midpoint is the origin. Then we have

for some . is not only , but it also depends on the x-coordinate we are observing. So

Next, we find the uniform linear charge density:

µ µ

and we can express the differential charge:

we are only interested in the y-component of the force, which is given by:

and is defined by

Now if we sub in , , and into our equation:

Now we integrate, noting that and are constants

Integration

We will use trig sub, setting

(see Pythagorean identities)

To return to , we draw a triangle. Since , then we have a base of , a height of , and a hypotenuse of . Therefore, .

plugging in our values , , and , we obtain

A positive point charge, , is located at a distance directly above the centre of a charged thin non-conducting circular plate of radius . The plate carries a total positive charge, , spread uniformly over its surface area. What will be the electrical force on the point charge?

solution

We calculate the surface charge density by dividing by the area:

Next, we divide the surface into differential area elements, .

The equation for the force is

Since the charge is placed at the centre of the circle, we only need to consider the z-component of the force, which is defined by: