Magnetic Fields and Forces

• Straight wire (right-hand rule 1). Point your right thumb along the conventional current; your curled fingers show the direction of the circular field lines around the wire. The field is strongest close to the wire and weakens with distance.
• Coil or solenoid (right-hand rule 2). Curl the fingers of your right hand in the direction of the conventional current in the loops; your thumb points out of the coil's north end. Inside a long solenoid the field lines are nearly parallel and evenly spaced, so the field there is close to uniform.
• Magnetic field lines always form closed loops. They leave the north pole of a magnet, enter the south pole and continue through the magnet. There are no isolated magnetic poles, unlike positive and negative charges.
• Field strength is measured in teslas (T). Earth's field at the surface is roughly $5\times 10^{-5}\text{T}$, while hospital MRI magnets commonly run at $1.5$ to $3\text{T}$.

The force is largest when the charge moves perpendicular to the field ($\theta =90^{\circ }$, $\sin \theta =1$) and zero when it moves parallel to the field. A stationary charge feels no magnetic force at all. The force is always perpendicular to both $\vec{v}$ and $\vec{B}$. To find its direction for a positive charge use right-hand rule 3: point your fingers along $\vec{B}$ and your thumb along $\vec{v}$; your palm pushes in the direction of the force. For a negative charge, such as an electron, find the direction for a positive charge and then reverse it. Rearranging $F=qvB$ also defines the tesla: $1\text{T}=1\text{N}/(\text{A}\cdot \text{m})$.

Because the magnetic force is always perpendicular to the velocity, it never does work: it changes the direction of motion but never the speed. A charge moving perpendicular to a uniform field therefore travels in a circle, with the magnetic force acting as the centripetal force. Setting $qvB=m{v}^2/r$ and solving for the radius gives the result below. Faster or heavier particles swing in wider circles; stronger fields or larger charges bend them more tightly.

A current is just many charges moving together, so a wire carrying current in a magnetic field feels the sum of all their magnetic forces. This is the motor principle from Grade 11 in equation form. The direction comes from right-hand rule 3 with your thumb along the conventional current. A wire running parallel to the field feels no force, and a wire perpendicular to it feels the maximum force $BIL$.

A mass spectrometer sorts ions by mass. The ions first pass through a velocity selector, a region with electric and magnetic fields at right angles, where the electric force $qE$ and magnetic force $qvB$ push in opposite directions. Only ions with $v=E/B$ feel balanced forces and pass straight through. These ions then enter a second uniform magnetic field and curve in semicircles of radius $r=mv/(qB)$. With speed, charge and field fixed, the radius is proportional to the mass, so heavier ions land farther away. Chemists use this to identify substances and to measure isotope abundances. A cyclotron uses a magnetic field to keep charged particles on a spiral path while an alternating electric field gives them a push every half-turn; hospital cyclotrons produce short-lived radioactive isotopes for medical imaging such as PET scans.