D.3 Motion in electromagnetic fieldsIB Physics SL: Revision notes

A charge q in a uniform electric field feels a constant force F = qE, in the direction of the field for positive charges and opposite to it for negative charges. Its acceleration is a = qE/m.

• Moving along the field lines, the particle speeds up or slows down in a straight line; the energy gained is qV.
• Entering perpendicular to the field, it keeps its original velocity at right angles to the field and accelerates uniformly along the field, so its path is a parabola, just like projectile motion. Use t = L/v for the time between plates, then y = ½at².

A charge moving through a magnetic field feels a force F = qvB sin θ, where θ is the angle between the velocity and the field. The force is maximum when v is perpendicular to B and zero when v is parallel to B.

The force is perpendicular to both v and B. Its direction is given by the right-hand rule (or Fleming's left-hand rule) for a positive charge; for a negative charge the force is reversed.

Because the magnetic force is always perpendicular to the velocity, it does no work: the speed and kinetic energy stay constant and only the direction changes. A particle moving perpendicular to a uniform field therefore moves in a circle, with the magnetic force as the centripetal force: qvB = mv²/r, so r = mv/(qB). Faster or heavier particles move in larger circles; a stronger field or larger charge gives a smaller circle. If v has a component along B, the path becomes a helix.

When uniform E and B fields are at right angles to each other and to the particle's velocity, the electric force qE and magnetic force qvB can act in opposite directions. The particle passes undeflected when qE = qvB, that is when v = E/B. This is a velocity selector: it lets through only particles of one speed, whatever their charge, sign or mass. Slower particles are deflected in the direction of the electric force; faster particles in the direction of the magnetic force.

A current is a flow of moving charges, so a conductor carrying a current I in a magnetic field feels a force F = BIL sin θ, where L is the length of conductor in the field and θ is the angle between the conductor and the field. The force is perpendicular to both the current and the field, and is greatest when they are perpendicular; there is no force when the conductor is parallel to the field.

Each current-carrying wire produces a circular magnetic field around it, and the other wire sits in that field, so each wire feels a force. The force per unit length is F/L = μ₀I₁I₂/(2πr), where r is the separation of the wires and μ₀ = 4π × 10⁻⁷ T m A⁻¹.

• Currents in the same direction attract.
• Currents in opposite directions repel.

The forces on the two wires are equal and opposite (Newton's third law), even if the currents are different.