Forces between multiple charges

The superposition principle

Narrated video. Transcript:

Coulomb’s law tells us the force between two charges. But what if there are three charges, q₁, q₂ and q₃? What is the total force on q₁?

Experiments give a simple answer. The force between any two charges is not changed by the presence of other charges. This is the principle of superposition.

So find the force on q₁ due to q₂ alone, F₁₂, using Coulomb’s law, as if q₃ were not there. Then find F₁₃, the force due to q₃ alone.

The total force on q₁ is the vector sum: F₁ equals F₁₂ plus F₁₃. Draw the two arrows from q₁ and complete the parallelogram. Its diagonal is the resultant.

For any number of charges, F₁ equals F₁₂ plus F₁₃ plus and so on up to F₁ₙ. Each term is k q₁ qᵢ over r₁ᵢ², along the line from qᵢ to q₁.

Example: three charges at the corners of an equilateral triangle of side l. Charges plus q at A and B, and minus q at C. Each pair is a distance l apart, so every force has the same size, F equals k q² over l².

On A: B pushes it away along BA, and C pulls it towards C. Two equal forces at 120° to each other. Their resultant also has size F, and it points parallel to BC.

On B, by the same argument, the resultant is F, parallel to AC. On C, A and B both pull it, with forces 60° apart. Their resultant is √3 F, pointing towards the middle of AB.

Add the three resultants, F along BC, F along AC and √3 F towards AB, and you get zero. The total force on the whole system is zero, as Newton’s third law demands.

Adding forces with components

Narrated video. Transcript:

A worked example. q₁ equals plus 1 microcoulomb sits at the origin. q₂ equals plus 3 microcoulombs is 3 centimetres away on the x-axis. q₃ equals minus 4 microcoulombs is 4 centimetres away on the y-axis.

Force due to q₂: 9 × 10⁹ times 1 × 10⁻⁶ times 3 × 10⁻⁶, over 0.03 squared. That is 0.027 over 0.0009, which is 30 newtons. Both are positive, so q₁ is pushed away from q₂: along minus x.

Force due to q₃: 9 × 10⁹ times 1 × 10⁻⁶ times 4 × 10⁻⁶, over 0.04 squared. That is 0.036 over 0.0016, which is 22.5 newtons. The signs are unlike, so q₁ is pulled towards q₃: along plus y.

The two forces are at right angles. The size of the resultant is the square root of 30 squared plus 22.5 squared: the square root of 1406.25, which is 37.5 newtons.

Its direction: tan α equals 22.5 over 30, which is 0.75, so α is about 37° above the minus x-axis. Measured from plus x, that is about 143°.

Notice that each force was worked out on its own, exactly as if the other charge did not exist. That is superposition.

Symmetry can save work. Put equal charges q at the three corners of an equilateral triangle, and a charge Q at the centroid. Each corner is the same distance from the centre.

So the three forces on Q are equal in size and 120° apart. They cancel: the net force on Q is zero. Remove one corner charge, and the force on Q is no longer zero.

Try it: three charges

Interactive simulation. Things to try:

• This is the worked example. The readouts give 30 N from q₂, 22.5 N from q₃, and a net force of 37.5 N at 143°.
• The dashed arrows are the forces from each charge alone. Drawn tip to tail, they end at the tip of the amber net force.
• Drag q₂ further away. Only F₁₂ changes; F₁₃ stays exactly the same. That is superposition.
• Click another charge to see the forces on it. Add a fourth charge with “+ charge”.

Charge in the middle

A +1 μC charge sits midway between a +2 μC charge and a −2 μC charge, which are 8 cm apart. What is the net force on the +1 μC charge?

1. Zero: the two forces cancel
2. 11.25 N towards the −2 μC charge
3. 22.5 N towards the −2 μC charge
4. 22.5 N towards the +2 μC charge

Triangle of charges

Charges +q, +q and −q sit at the corners of an equilateral triangle of side l. With F = kq²/l², what is the size of the net force on the −q charge?

1. Zero
2. F
3. √3 F
4. 2F

Summary

• Superposition: F₁ = F₁₂ + F₁₃ + … + F₁ₙ
• Each term: F₁ᵢ = k q₁qᵢ r̂₁ᵢ / r₁ᵢ²
• Two equal forces at angle θ: R = 2F cos(θ/2)

• The force between two charges is not affected by other charges nearby.
• The net force on a charge is the vector sum of the Coulomb forces from all the other charges, each worked out on its own.
• Add the forces with the parallelogram law or with x and y components.
• Use symmetry: equal forces spaced evenly round a point cancel.