understand that a magnetic field is an example of a field of force produced either by moving charges or by permanent magnets

Concept of a Magnetic Field

A magnetic field is a vector field that exerts a force on moving electric charges and on magnetic dipoles. It is produced by two main sources:

• Moving electric charges (currents)
• Permanent magnets (aligned magnetic moments)

1. Definition and Representation

The magnetic field at a point in space is represented by the vector 𝐁. Its magnitude is measured in teslas (T) and its direction is given by the right-hand rule for currents or the orientation of magnetic dipoles.

Mathematically, the force on a charge q moving with velocity 𝐯 in a magnetic field 𝐁 is:

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= q (𝐯 × 𝐁)

where “×” denotes the cross product. The force is perpendicular to both 𝐯 and 𝐁.

2. Sources of Magnetic Fields

2.1 Moving Charges (Currents)

According to the Biot–Savart law, a small segment of current I flowing through a differential length dl produces a magnetic field at point r given by:

\$\mathrm{d}\mathbf{B} = \frac{\mu_0}{4\pi}\frac{I\,\mathrm{d}\mathbf{l}\times\hat{\mathbf{r}}}{r^2}\$

Integrating over the entire current path yields the total magnetic field.

2.2 Permanent Magnets

Permanent magnets consist of many microscopic magnetic dipoles m aligned in the same direction. The magnetic field produced by a dipole at a point in space is:

\$\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi}\frac{3(\mathbf{m}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{m}}{r^3}\$

where r is the distance from the dipole and hat{r} is the unit vector pointing from the dipole to the field point.

3. Key Properties of Magnetic Fields

1. Field lines are continuous. They never begin or end; they form closed loops.
2. Direction of field lines. For a bar magnet, they emerge from the north pole and re-enter at the south pole.
3. Superposition principle. The net magnetic field is the vector sum of all individual contributions.

4. Magnetic Field of a Straight Current-Carrying Wire

For an infinitely long straight wire carrying current I, the magnetic field at a perpendicular distance r is:

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

Its direction is given by the right-hand rule: if the thumb points along the current, the curled fingers show the direction of 𝐁.

5. Ampère’s Law

For a closed loop C in a static magnetic field, Ampère’s law states:

\$\ointC \mathbf{B}\cdot\mathrm{d}\mathbf{l} = \mu0 I_{\text{enc}}\$

where I_enc is the net current passing through the area bounded by C.

6. Summary Table: Field Sources and Their Characteristics

7. Practical Examples

• Deflection of a charged particle beam. In a cathode ray tube, electrons are steered by a magnetic field.
• Magnetic levitation. A superconducting magnet creates a field that counteracts gravity on a levitating object.
• Electromagnetic induction. A changing magnetic field induces an electromotive force in a coil (Faraday’s law).

8. Key Takeaways

1. A magnetic field is a force field that acts on moving charges and magnetic dipoles.
2. It can be produced by electric currents or by permanent magnets.
3. Its magnitude and direction are described by vector equations such as the Biot–Savart law and Ampère’s law.
4. Field lines are continuous and form closed loops, never beginning or ending.