A-level Physics/Forces, Fields and Energy/Electric fields

Like gravitational fields, electric fields are a field of force that act from a distance, where the force here is exerted by a charged object on another charged object. You may already be familiar with the fact that opposite charges attract, and that like charges repel. Here, we will look at ways to calculate field strengths and the magnitude of forces exterted, in a very similar manner to gravitational fields.

Representing electric fields

Electric field lines are drawn always pointing from positive to negative, like the flow of current. Just like magnetic and gravitational fields, the separation of the lines tell us the relative strength.

Radial fields

Radial fields are drawn from a centre point. The field is stronger nearer the surface of the object, and weakens as you move further away. For a positive charge, the arrows point outwards, and for a negative charge, the arrows point inwards.

Uniform fields

For a uniform electric field between two charged plates, the lines are drawn perfectly parallel, from positive to negative. The field curves slightly outwards on the edge of the plates, and you should remember to draw it like that. However, this curvature is usually ignored for calculations. Well inside the plates the electric field is uniform, constant strength from one plate to the other.

Multiple charges

When there are several radial and uniform fields close to each other, they have to be combined into one field, since each of their fields interact and change. The most common shapes are shown, and the arrows, as always, point from positive to negative. You should be able to draw field lines for simple variations on these.

Coulomb's law

Coulomb's law is very similar to Newton's law of gravitation, except instead of relating the force between two masses together, it relates the force between two charges, $Q_1$ and $Q_2$. Since the two charges are point charges which have radial fields, they follow the inverse square law.

Therfore, the relationship can be expressed as:

$F\propto \frac{Q_1{Q}_2}{r^2}$.

Or, in words:

Any two point charges exert a force on each other that is proportional to the product of their charges and inversely proportional to the square of the distance between them.

Just like Newton's law, we need to introduce a constant of proportionality to make it into an equation, which in this case is k:

$F=k\frac{Q_1{Q}_2}{r^2}$.

Where $k=\frac{1}{4\pi {ϵ}_0}\approx 8.99\times 10^9$.

Permittivity of free space

${ϵ}_0$ is known as the permittivity of free space, and is roughly $8.85\times 10^{-12}$. It is often useful to just remember that $k\approx 8.99\times 10^9$ in free space, however you do also need to know $k=\frac{1}{4\pi {ϵ}_0}$, as you may be given the permittivity of different mediums.

Signs of charges

Note that for each charge, you must keep the signs intact in the equation. If you were to have two positive, or two negative charges in the equation, the result would be positive, but if you were to have one negative and one positive charge, the final answer would be negative. The sign of the answer tells us whether the force between the two charges is an attraction, or a repulsion, like charges will repel, and opposite charges will attract. This also explains the minus sign in Newton's law of gravitation, since the force between two masses is always an attraction.

Electric field strength

Just as gravitational field strength is the force exerted per unit mass, we could define the electric field strength in terms of charge:

The electric field strength at a point is the force per unit charge exerted on a positive charge placed at that point.

This is just like saying that the electric field strength is the force a charge of +1 coulomb experiences in that electric field. Therfore, we can find the electric field strength, E, by:

$E=\frac{F}{Q}$.

From this equation, you can see that the electric field strength is measured in $N{C}^{-1}$.

Field strength of a uniform field

You can make a uniform electric field by charging two plates. Increasing the voltage between them will increase the field strength, and moving the plates further apart will decrease the field strength. A simple equation for field strength can be made from these two points:

$E=-\frac{V}{d}$

Where V is the voltage between the plates, and d is the distance between them. Note the minus sign in the equation, which has been added since the force that a positive charge will experience in the field is away from the positively charged plate.

Here you can see that the units of electric field strength is $V{m}^{-1}$. $N{C}^{-1}$ is equivalent to $V{m}^{-1}$.

Field strength of a radial field

Since the electric field strength could be said to be the force exerted on a charge of +1C, we can substitute 1 coulomb for $Q_2$ in Coulomb's law. We then get the equation:

$E=\frac{kQ}{r^2}$, or

$E=\frac{Q}{4\pi {ϵ}_0{r}^2}$

This will tell us the field strength of a charge, Q, at a distance, r.

Force on particles

To calculate the force an electron experiences in a uniform field, we can combine $E=\frac{F}{Q}$ with $E=-\frac{V}{d}$ in the following steps:

$\frac{F}{Q}=-\frac{V}{d}$

$F=-\frac{QV}{d}$

For an electron with a charge of -e, this becomes:

$F=\frac{eV}{d}$, or $eE$

This is useful if you are asked to find the force on an electron in a uniform field, most often in a cathode ray tube.

Electric Potential and Work

Electric Potential is defined as the work that must be done to bring unit charge from infinity to a point in an electric field, a distance r from the center of the charge causing the field.

$V$	$=\frac{1}{4\pi {ϵ}_0}\times \frac{Q}{r}$
	$=\frac{Q}{4\pi {ϵ}_{0}r}$

The electric field strength at the point where the unit charge is, is equal to the negative potential gradient at that point.

$E$	$=-\frac{dV}{dr}$
	$=-\frac{d}{dr}\left(\frac{Q}{4\pi {ϵ}_{0}r}\right)$
	$=-\frac{Q}{4\pi {ϵ}_0}\times \frac{d}{dr}\left(\frac{1}{r}\right)$
	$=\frac{Q}{4\pi {ϵ}_0}\times \frac{1}{r^2}$
	$=\frac{Q}{4\pi {ϵ}_0{r}^2}$

Electric Potential Energy is defined as the work that must be done to bring a specific charge from infinity to a point in an electric field, a distance r from the center of the charge causing the field.

$U$	$=\frac{1}{4\pi {ϵ}_0}\times \frac{Q_1{Q}_2}{r}$
	$=\frac{Q_1{Q}_2}{4\pi {ϵ}_{0}r}$
	$=Q_{2}V$

The electric force acting on the charge is equal to the negative potential energy gradient at that point.

$F$	$=-\frac{dU}{dr}$
	$=-\frac{d}{dr}\left(\frac{Q_1{Q}_2}{4\pi {ϵ}_{0}r}\right)$
	$=-\frac{Q_1{Q}_2}{4\pi {ϵ}_0}\times \frac{d}{dr}\left(\frac{1}{r}\right)$
	$=\frac{Q_1{Q}_2}{4\pi {ϵ}_0}\times \frac{1}{r^2}$
	$=\frac{Q_1{Q}_2}{4\pi {ϵ}_0{r}^2}$

Comparison of electric and gravitational fields

As you may have already noticed, electric and gravitational fields are quite similar. You should be aware of the similarities and differences between them.

Similarities

• For point charges or masses, the variation of force with distance follows the inverse square law.
• Both exert a force from a distance, with no contact.
• The field strength of both is defined in terms of force per unit of the property of the object that causes the force (i.e. mass and charge).

Differences

• Gravitational fields can only produce forces of attraction, wheras electric fields can produce attraction and repulsion.
• Objects can be shielded from an electric field, they cannot however be shielded from a gravitational field