Magnetic Field

1. Introduction

A magnetic field is the region around a magnet or a current-carrying conductor in which a magnetic force can be experienced. The Danish physicist Oersted discovered that an electric current produces a magnetic field, linking electricity and magnetism. This chapter studies magnetic field lines and flux, the force on moving charges and currents, the torque on a coil (and its use in the galvanometer), the Hall effect, and the laws (Biot-Savart and Ampere) that give the field produced by currents.

2. Magnetic Field Lines and Magnetic Flux (Subtopic 16.1)

A magnetic field line is a line such that the tangent drawn at any point on it gives the direction of the magnetic field at that point. Field lines are continuous closed loops (they have no beginning or end), they never cross, and their closeness indicates the field strength.

• Around a long straight current-carrying conductor, the field lines are concentric circles in planes perpendicular to the wire; their direction is given by the right-hand grip rule (thumb along the current, curled fingers give the field).
• Inside a long solenoid the field lines are straight, parallel, and equally spaced (a nearly uniform field), emerging from one end (north pole) and looping round to the other, resembling a bar magnet.

Magnetic flux $\Phi$ through a surface of area $A$ in a uniform field $B$ making angle $\theta$ with the normal to the area is:

$\Phi = BA\cos\theta$

Its SI unit is the weber (Wb); the field $B$ (flux density) is measured in $\text{tesla (T)} = \text{Wb m}^{-2}$.

3. Oersted's Experiment (Subtopic 16.2)

Oersted's experiment (1820): a magnetic compass needle placed near a straight wire is deflected when a current flows through the wire, and the deflection reverses when the current is reversed.

Outcome: a current-carrying conductor produces a magnetic field around it; the field is circular around the wire, and its direction depends on the direction of the current.

Limitations: the experiment shows only that a current produces a magnetic field and gives its direction; it does not give the magnitude of the field, nor does it work for very small currents (the deflection is too small to detect).

4. Force on a Moving Charge in a Magnetic Field (Subtopic 16.3)

When a charge $q$ moves with velocity $v$ through a magnetic field $B$, making an angle $\theta$ with the field, it experiences a force (the Lorentz force). Experiment shows this force is proportional to $q$, to $v$, to $B$, and to $\sin\theta$:

$F = qvB\sin\theta$

In vector form:

$\vec{F} = q(\vec{v}\times\vec{B})$

The force is always perpendicular to both $\vec{v}$ and $\vec{B}$.

• When $\theta = 0^\circ$ or $180^\circ$ (charge moving parallel to the field), $F = 0$.
• When $\theta = 90^\circ$, the force is maximum, $F_{\max} = qvB$.

Because the force is perpendicular to the velocity, it does no work and only changes the direction of motion; a charge entering perpendicular to a uniform field therefore moves in a circle of radius $r = \dfrac{mv}{qB}$.

5. Force on a Current-Carrying Conductor (Subtopic 16.4)

Consider a conductor of length $l$ and cross-sectional area $A$ placed at angle $\theta$ in a uniform field $B$, carrying current $I$. If $n$ is the number of free electrons per unit volume, each moving with drift velocity $v_d$, the force on one electron is $F_1 = ev_d B\sin\theta$. The total number of electrons in the conductor is $N = nAl$, so the total force is:

$F = (nAl)\,e v_d B\sin\theta$

Since the current is $I = neAv_d$, this becomes:

$\boxed{F = BIl\sin\theta}$

In vector form $\vec{F} = I(\vec{l}\times\vec{B})$.

• When $\theta = 0^\circ$ (conductor parallel to $B$): $F = 0$.
• When $\theta = 90^\circ$: $F_{\max} = BIl$.

The direction of the force is given by Fleming's left-hand rule: stretch the thumb, forefinger, and middle finger of the left hand mutually perpendicular; the forefinger points along the field, the middle finger along the current, and the thumb gives the force (direction of motion).