Lesson 3 of 6Gravitational, Electric, and Magnetic Fields

Unit 3 · Lesson 3 of 6

Electric Forces and Fields

Coulomb's law for the force between charges, how to add several electric forces as vectors, and the electric field that a charge sets up around itself.

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Key things to remember

1Coulomb's law: $F_{e} = \frac{k q _{1} q _{2}}{r ^{2}}$ with $k = 8.99 \times 1 0^{9} N \cdot m^{2} / C^{2}$ . Like charges repel; unlike charges attract.
2Superposition: the net force (or field) from several charges is the vector sum of the individual forces (or fields).
3Electric field strength: $E = F_{e} / q$ in N/C, directed as the force on a positive test charge.
4Field of a point charge: $E = \frac{k q}{r ^{2}}$ , pointing away from positive and toward negative charges.
5Between large parallel plates the field is uniform; around a point charge it is radial and inverse square.

In earlier science courses you learned that like charges repel, unlike charges attract, and that charge comes in units of the elementary charge $e = 1.60 \times 1 0^{- 19} C$ . This topic makes that quantitative. The electric force turns out to follow almost exactly the same pattern as gravity, an inverse square law, but it can push as well as pull, and it is staggeringly stronger. It is the force that holds atoms, molecules and every solid object together.

Coulomb's law

F_{e} = \frac{k q _{1} q _{2}}{r ^{2}} k = 8.99 \times 1 0^{9} N \cdot m^{2} / C^{2}

Charges in coulombs, r in metres between the centres of the charges. Use the magnitudes of the charges to find the size of the force.

ACoulomb's law (electric)

$F_{e} = k q_{1} q_{2} / r^{2}$ , an inverse square law
Acts between charges
Attractive or repulsive, depending on the signs
$k$ is huge, so the force is strong even for small charges
Can be shielded or cancelled by opposite charges

BUniversal gravitation

$F_{g} = G m_{1} m_{2} / r^{2}$ , an inverse square law
Acts between masses
Always attractive
$G$ is tiny, so the force is weak unless a mass is enormous
Cannot be shielded; every mass adds to it

Magnitude first, direction second

Substitute the sizes of the charges into Coulomb's law to get the size of the force. Then decide the direction from the physics: like charges push each other apart, unlike charges pull together. Plugging in negative signs and trusting the sign of the answer causes errors as soon as more than two charges are involved, because a negative answer does not tell you which way along the line to point.

Superposition: more than two charges

When several charges act on one charge, each pair interacts as if the others were not there. The net force is the vector sum of the individual forces; this is the principle of superposition. In one dimension, choose a positive direction, give each force a sign by its direction, and add. In two dimensions, resolve each force into components, add the components, then rebuild the resultant with the Pythagorean theorem and trigonometry.

Sketch the charges and mark the one you are finding the force on.
Find the size of each force on that charge with Coulomb's law, one pair at a time, using magnitudes.
Draw each force from the charge, pointing away from like charges and toward unlike charges.
Resolve into components with a stated positive direction for each axis, then add the components.
Rebuild the resultant: magnitude from $F_{x}^{2} + F_{y}^{2}$ and direction from $tan^{- 1}$ , written as a bearing such as [E 16° S].

The electric field

Just as a mass creates a gravitational field, a charge creates an electric field in the space around it. The electric field strength $E$ at a point is the electric force per coulomb on a small positive test charge placed there, $E = F_{e} / q$ , measured in newtons per coulomb. Its direction is defined as the direction of the force on a positive test charge, so a negative charge placed in the field feels a force opposite to $E$ . Dividing Coulomb's law by the test charge gives the field of a single point charge $q$ .

E = \frac{F _{e}}{q} E = \frac{k q}{r ^{2}} F_{e} = q E

The field points away from a positive source charge and toward a negative one. Fields from several charges add as vectors.

Fig 1Field lines for a positive charge, a negative charge, a dipole, and the uniform field between parallel plates.

Field lines start on positive charges and end on negative charges, and an arrow on each line shows the direction a positive test charge would be pushed.
The closer the lines, the stronger the field. Around a point charge they spread out radially, which is the inverse square law made visible.
Field lines never cross. At any point the field has one direction only.
Lines meet a conductor's surface at right angles, which is why the lines leave charged plates perpendicular to them.
The number of lines is proportional to the charge, so a charge of $- 2 q$ has twice as many lines ending on it as $+ q$ has leaving.

Point charges versus parallel plates

Around a point charge the field is radial and weakens as

1/ r^{2}

. Between two large, oppositely charged parallel plates the lines are straight, parallel and evenly spaced, so the field is uniform: the same size and direction everywhere between the plates, except near the edges. This mirrors gravity, where the field is radial far from Earth but effectively uniform near the surface. The next topic uses the uniform field between plates to accelerate charges.

How much stronger is the electric force

Between the proton and electron in a hydrogen atom, the ratio of the electric attraction to the gravitational attraction is

\frac{k e ^{2}}{G m _{e} m _{p}} \approx 2.3 \times 1 0^{39}

, independent of the distance because both forces are inverse square. Gravity still dominates on the scale of planets because matter is almost perfectly neutral overall, so electric forces cancel out while every kilogram adds to gravity.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

+ 3.00 μ C

charge and a

- 5.00 μ C

charge are

12.0 cm

apart. Find the electric force between them.

Example 2

Three charges lie on the x-axis:

q_{1} = + 2.0 μ C

x = 0

q_{3} = + 1.0 μ C

x = 0.10 m

, and

q_{2} = - 4.0 μ C

x = 0.30 m

. Find the net electric force on

q_{3}

. Take right (

+ x

) as positive.

Example 3

+ 2.0 μ C

charge sits at the origin. A

+ 3.0 μ C

charge is

0.40 m

due north of it and a

- 6.0 μ C

charge is

0.30 m

due east of it. Find the net force on the charge at the origin.

Example 4

+ 5.0 nC

charge and a

- 3.0 nC

charge are

20 cm

apart. Find the electric field at the midpoint between them, and the force on an electron placed there.

Where marks get lost

Forgetting to convert microcoulombs ( $1 0^{- 6}$ ), nanocoulombs ( $1 0^{- 9}$ ) or centimetres before substituting. One missed conversion changes the answer by factors of a thousand or more.
Using the signs of the charges to set the direction of the force in a multi-charge problem. Find magnitudes, then decide each direction from attract or repel.
Adding the magnitudes of forces that point in different directions. Forces and fields are vectors; add components.
Thinking the field direction is the direction any charge moves. $E$ is defined by the force on a positive test charge; electrons are pushed the opposite way.
Including the test charge's own field when finding the field at its location. The field at a point comes from the other charges only.

Flashcards 8

Coulomb's law

Principle of superposition

Electric field strength

Field of a point charge

+ 4 more

Practice

Two small spheres carry charges of

+ 1.5 μ C

and

+ 2.5 μ C

and are

5.0 cm

apart. What is the size of the electric force between them, in newtons? Give three significant figures.

Both charges in a pair are doubled and the distance between them is also doubled. What happens to the electric force?

What is the magnitude of the electric field

25 cm

from a

- 8.0 nC

point charge, in N/C? Give three significant figures.

N/C

Which way do the electric field lines point near an isolated negative point charge?

3.0 μ C

charge placed at a point experiences an electric force of

0.060 N

. What is the electric field strength at that point, in N/C?

N/C

+ 4.0 μ C

charge is at

x = 0

and a

+ 1.0 μ C

charge is at

x = 30 cm

. At what position between them, in centimetres from the

+ 4.0 μ C

charge, is the net electric field zero?

What is the name of the principle that says the net force on a charge is the vector sum of the forces from each other charge acting alone?

In a hydrogen atom, why can the gravitational force between the proton and the electron be ignored?

Match each arrangement to the field it produces.

Isolated positive point charge

Isolated negative point charge

Two large oppositely charged parallel plates

A positive and a negative charge close together

Order the steps for finding the net electric force on a charge due to two other charges that are not in a line.

1Find the resultant's magnitude and direction from the component sums
2Resolve the forces into x- and y-components and add them
3Decide each force's direction from attraction or repulsion
4Convert all charges and distances to coulombs and metres
5Find the size of each force with Coulomb's law using magnitudes

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