represent a magnetic field by field lines

Concept of a Magnetic Field

Learning Objective

Students will be able to represent a magnetic field using magnetic field lines and understand the qualitative information conveyed by those lines.

1. What is a Magnetic Field?

A magnetic field, denoted by the vector symbol $\mathbf{B}$, is a region of space around a moving charge or a magnet within which other moving charges experience a magnetic force. The magnetic force on a charge $q$ moving with velocity $\mathbf{v}$ is given by

$$\mathbf{F}=q\,\mathbf{v}\times\mathbf{B}$$

The direction of $\mathbf{B}$ at any point is defined as the direction of the force that would act on a positive test charge moving with velocity $\mathbf{v}$ perpendicular to the field.

2. Magnetic Field Lines – Definition

Magnetic field lines are a visual tool used to illustrate the direction and relative strength of a magnetic field.

• They are drawn so that the tangent at any point gives the direction of $\mathbf{B}$.
• The density of lines (how close they are together) indicates the magnitude of the field: closer lines → stronger field.
• Field lines form continuous closed loops; they never start or end in space.
• Inside a magnet, field lines run from the south pole to the north pole; outside the magnet they run from north to south.

3. Rules for Drawing Magnetic Field Lines

1. Draw lines emerging from the north pole and entering the south pole of a magnet.
2. Outside the magnet, lines go from north to south; inside they return from south to north.
3. Lines never intersect.
4. The number of lines is proportional to the strength of the source (e.g., current or magnetic moment).
5. Use the right‑hand rule for current‑carrying conductors to determine the direction of the field.

4. Right‑Hand Rule for a Straight Current‑Carrying Wire

Place the thumb of the right hand in the direction of the conventional current $I$. The curled fingers then point in the direction of the magnetic field lines encircling the wire.

The magnitude of the field at a distance $r$ from a long straight wire is

$$B = \frac{\mu_0 I}{2\pi r}$$

5. Comparison of Magnetic Field Lines for Different Sources

Source	Field‑Line Pattern	Key Features
Bar Magnet	Closed loops from north to south outside the magnet	Uniform inside, denser near poles
Straight Current‑Carrying Wire	Concentric circles around the wire	Field strength $\propto 1/r$
Solenoid (long coil)	Parallel lines inside, spreading out at the ends	Inside field nearly uniform; $B = \mu_0 n I$
Earth	Approximately dipole field, lines emerging near the geographic south pole and entering near the north pole	Field strength varies with latitude

6. Using Field Lines to Predict Forces

When a current‑carrying conductor is placed in an external magnetic field, the direction of the force on the conductor can be found using the right‑hand rule for the cross product $\mathbf{F}=I\mathbf{L}\times\mathbf{B}$.

7. Summary

• Magnetic field lines are a convenient way to visualise the direction and relative magnitude of $\mathbf{B}$.
• They always form closed loops and never intersect.
• The density of lines indicates field strength.
• Right‑hand rules help determine the direction of $\mathbf{B}$ for currents and the direction of forces on moving charges.

8. Practice Questions

1. Draw the magnetic field lines for a bar magnet and label the north and south poles.
2. A straight wire carries a current of 5 A. Sketch the magnetic field lines at a distance of 2 cm from the wire.
3. Explain why magnetic field lines never start or end in empty space.
4. Using the right‑hand rule, determine the direction of the magnetic field at a point directly above the centre of a circular loop carrying clockwise current when viewed from above.
5. Two parallel wires carry currents in the same direction. Using field‑line concepts, explain why the wires attract each other.