Coulomb's law, electric fields, electric potential, magnetic fields and the right-hand rule

Coulomb's law is Newton's gravitation with a sign: F = kq₁q₂/r², with k = 8.99 × 10⁹ N·m²/C², attractive for opposite charges and repulsive for like ones. An electric field is that force per coulomb, E = F/q, pointing the way a positive test charge would be pushed. A magnetic field pushes only on a moving charge, F = qvB sin θ, at right angles to both the velocity and the field, and the right-hand rule names which way. Below, one Coulomb problem worked line by line, the field and the potential that come from it, and the proton in a magnetic field with the three errors students make.

Coulomb's law: gravitation with a sign

The Coulomb's law formula is F = kq₁q₂/r², where q₁ and q₂ are the two charges in coulombs, r is the distance between their centres in metres, and k = 8.99 × 10⁹ N·m²/C². It has the same shape as gravitation, the same inverse square: a product on top, a distance squared underneath, a constant in front. The one difference is that charge comes in two signs and mass does not, so the electric force can push as well as pull.

The working habit: compute the size of the force with the absolute values of the charges, then read the direction off the signs. Opposite signs attract, each charge pulled toward the other. Like signs repel, each pushed away. Do not try to carry the sign through the arithmetic; it tells you a direction, and you draw directions.

Two unit conversions break more Coulomb lines than anything else in the formula. Charges arrive in microcoulombs: 1 μC = 10⁻⁶ C. Distances arrive in centimetres: 1 cm = 10⁻² m. Both go into SI before the first line, and r is squared after the conversion, never before.

Worked example: +2.0 μC and −3.0 μC, 5.0 cm apart

Two point charges, +2.0 μC and −3.0 μC, sit 5.0 cm apart. Find the force between them, and the electric field the first charge produces at the second's position.

Givens, converted: q₁ = +2.0 × 10⁻⁶ C, q₂ = −3.0 × 10⁻⁶ C, r = 0.050 m, k = 8.99 × 10⁹ N·m²/C². The data carry two significant figures, so the answers do; the lines keep three.

Line 1, the distance squared.

• r² = (0.050 m)² = 2.5 × 10⁻³ m² (squaring after the conversion to metres)

Line 2, the product of the charges, as sizes.

• |q₁q₂| = 2.0 × 10⁻⁶ C × 3.0 × 10⁻⁶ C = 6.0 × 10⁻¹² C² (multiplying the magnitudes; the signs are kept for the direction)

Line 3, the force.

• F = k|q₁q₂|/r² = 8.99 × 10⁹ N·m²/C² × 6.0 × 10⁻¹² C² ÷ 2.5 × 10⁻³ m² (the formula with the numbers in)
• F = 5.39 × 10⁻² N·m² ÷ 2.5 × 10⁻³ m² = 21.6 N, which is 22 N to two figures (the C² cancel, the m² cancel, newtons remain)
• Direction: the signs are opposite, so the force is attractive; each charge is pulled toward the other with 22 N

Line 4, the field q₁ makes where q₂ sits. The field of a point charge is E = k|q|/r², the force per coulomb it would exert on a test charge there.

• E = 8.99 × 10⁹ N·m²/C² × 2.0 × 10⁻⁶ C ÷ 2.5 × 10⁻³ m² = 7.2 × 10⁶ N/C (one C cancels, leaving N/C)
• Direction: away from q₁, because q₁ is positive and a field points the way a positive test charge would be pushed

The check that teaches what a field is. Put q₂ in that field: F = |q₂|E = 3.0 × 10⁻⁶ C × 7.2 × 10⁶ N/C = 21.6 N, the same force as line 3. The field is Coulomb's law with one charge taken out, waiting for whatever charge is placed there. Two routes to one number is the pattern every explainer on this site uses as its check; getting the same 21.6 N both ways is how you know lines 1 to 4 hold.

The electric field: force per coulomb, and how to add two of them

E = F/q, in newtons per coulomb. A positive charge's field points away from it in every direction; a negative charge's field points toward it. The field is a vector, so when two charges both make a field at the same point, the two fields add as vectors: by components, head to tail, never by adding their sizes unless they happen to point the same way. Two equal positive charges make zero field at the midpoint between them, because the two fields there are equal in size and opposite in direction.

Field lines are the picture of this. They start on positive charges and end on negative ones; where they crowd together the field is strong; the direction of a line at a point is the direction a positive test charge would be pushed there.

Electric potential: energy per coulomb

The electric potential of a point charge is V = kq/r, in volts, and a volt is a joule per coulomb. Where the field was force per coulomb, the potential is energy per coulomb. The potential is a scalar: it carries the sign of its charge, but it has no direction, so the potentials of two charges at a point add as plain numbers. That is the easiest way to tell the two quantities apart on a test. Fields are added by components; potentials are added up.

A potential difference ΔV between two points is the work per coulomb to move a charge between them, so the work on a charge q is W = qΔV. A positive charge moves on its own from high potential to low, downhill in V; a negative charge moves uphill. Between two parallel plates the field is uniform and E = ΔV/d, which gives the field a second unit, volts per metre. The two units are the same thing: V/m = (J/C)/m = (N·m/C)/m = N/C.

One number for the scale. An electron crossing a potential difference of 500 V gains kinetic energy W = qΔV = 1.60 × 10⁻¹⁹ C × 500 V = 8.0 × 10⁻¹⁷ J. Written in electron volts that is simply 500 eV, which is why the eV exists as a unit.

Magnetic fields push only on moving charges

A magnetic field, measured in teslas, exerts no force on a charge at rest, and none on a charge moving along the field. On a charge moving across it the force is F = qvB sin θ, where θ is the angle between the velocity and the field. The force is at right angles to both v and B, which means it is never along the motion, so it does no work and never changes the speed. It only turns the velocity.

A charge moving at right angles to a uniform field is therefore turned round and round in a circle. Setting the magnetic force equal to the centripetal requirement, qvB = mv²/r, gives the radius r = mv/(qB): the circle, from mv²/r once more, with magnetism as the supplier.

A proton moving at 2.0 × 10⁶ m/s perpendicular to a 0.50 T field, with e = 1.60 × 10⁻¹⁹ C and mass 1.67 × 10⁻²⁷ kg:

• F = qvB = 1.60 × 10⁻¹⁹ C × 2.0 × 10⁶ m/s × 0.50 T = 1.6 × 10⁻¹³ N (sin 90° = 1; C × m/s × T = N)
• r = mv/(qB) = 1.67 × 10⁻²⁷ kg × 2.0 × 10⁶ m/s ÷ (1.60 × 10⁻¹⁹ C × 0.50 T) (the radius formula with the numbers in)
• r = 3.34 × 10⁻²¹ ÷ 8.0 × 10⁻²⁰ = 0.042 m, which is 4.2 cm (the division)

On a wire carrying a current I over a length L, the moving charges add up to F = BIL sin θ, with θ the angle between the wire and the field.

The right-hand rule, stated once and used twice

The right hand rule, as this page uses it: for a positive charge, point the fingers of your right hand along the velocity, curl them toward the magnetic field, and your thumb points along the force. There is a second form taught in some classrooms, with the fingers along B, the thumb along v and the palm pushing in the direction of F. It is the same rule with the hand held differently and gives the same answers; pick one and use it for the whole course.

Two additions. For a negative charge, the rule gives the force on a positive charge, so reverse the answer. For a wire, the thumb or the fingers go along the current, since current is the direction positive charge moves.

Applied to the proton above, with the field into the page and the proton moving to the right: fingers to the right along v, curl them into the page toward B, and the thumb points up the page. The force on the proton is up the page, 1.6 × 10⁻¹³ N, and the proton curves upward into a circle of radius 4.2 cm. An electron moving the same way would curve downward.

The four wrong lines a marker sees

1. r not squared, or squared before the cm → m conversion. Squaring 5.0 cm as 25 and then converting to 0.25 m² is wrong by a factor of 100; forgetting to convert at all and squaring 5.0 gives a denominator 10⁴ times too large, and a force 10⁴ times too small, 2.2 × 10⁻³ N here. The symptom is a power of ten that does not match a quick estimate.
2. μC left as a bare number. Putting 2.0 and 3.0 into the formula instead of 2.0 × 10⁻⁶ and 3.0 × 10⁻⁶ gives a force 10¹² times too large. The symptom is a number no two small charges could produce.
3. The right-hand rule applied to an electron without reversing. The rule is written for positive charge. The symptom is a particle curving the wrong way in a diagram the question has already drawn.
4. Two electric fields added as magnitudes when they point different ways, or two potentials added as vectors. Fields are vectors, potentials are scalars. The symptom in the first case is a field at the midpoint between two like charges that should be zero and is not.

A fifth: F = qvB used for a charge moving along the field. sin 0° = 0, so the force is zero; a charge moving along B feels nothing.

A sibling to try cold

Cover the example. Two charges, +4.0 μC and +1.0 μC, are 20 cm apart. Find the force between them and its direction, and the field of the 4.0 μC charge at the other charge's position. Convert both units before any line, and square r only after. Redo it cold, then a sibling is the pattern: the example above was the demonstration, this is the attempt, and Sweller and Cooper (1985) found that the pair teaches a procedure faster than attempts alone.

Check

• r = 0.20 m, r² = 0.040 m²
• F = 8.99 × 10⁹ × 4.0 × 10⁻⁶ × 1.0 × 10⁻⁶ ÷ 0.040 = 8.99 × 10⁹ × 4.0 × 10⁻¹² ÷ 0.040 = 0.90 N, repulsive (both positive; each pushed away from the other)
• E = 8.99 × 10⁹ × 4.0 × 10⁻⁶ ÷ 0.040 = 9.0 × 10⁵ N/C, pointing away from the 4.0 μC charge
• Check: F = 1.0 × 10⁻⁶ C × 9.0 × 10⁵ N/C = 0.90 N, the same

Where this sits in the course

This page covers ideas 16, 17 and 18 of 25 in SPH4U, the second half of strand D: Coulomb's law and electric fields, Electric potential and potential difference and Magnetic fields and the force on moving charges and currents. Before them is idea 15, Gravitational potential energy and potential, which has the same 1/r shape as V = kq/r; the course teaches Coulomb as gravitation with a sign on purpose, and potential energy and the 1/r curve is where that shape first appears.

After them the light strand opens with idea 19, Interference of light; Young's double slit. The light strand, the idea after this one is written the same way as this page. The SPH4U course map shows all 25 ideas in teaching order, and every idea of the three science courses has a page like this one.

Fields on one screen

• Two force laws. F = kq₁q₂/r² between charges; F = qvB sin θ on a moving charge. Both computed with magnitudes, the direction drawn afterwards.
• Field and potential. E = F/q = kq/r², a vector in N/C, added by components. V = kq/r, a scalar in volts, added as numbers. E = ΔV/d between plates; V/m = N/C.
• Magnetism. No force on a charge at rest or moving along B; at right angles, a circle of radius r = mv/(qB), the speed unchanged. On a wire, F = BIL sin θ.
• The right-hand rule. Fingers along v, curl toward B, thumb along F, for a positive charge; reverse it for a negative one.
• The four wrong lines. r unsquared or squared before converting; μC left bare; the rule unreversed for an electron; fields added as sizes or potentials as vectors.
• The habit. Convert first, derive the field from the force every time, and check F two ways; how to study physics is built on that.