Magnetic Effects of Current & Magnetism
(INSPIRE Scholarship – Physics – Study Notes)
1. Simple Explanation
When an electric current flows through a conductor, it creates a magnetic field around it.
- Direction – given by the right‑hand rule (thumb along the current, fingers curl in the direction of the field).
- Magnitude – proportional to the current and depends on the geometry of the conductor.
- Interaction – two currents exert forces on each other; a current loop in a magnetic field experiences a torque.
- Materials – some materials (ferromagnets, paramagnets, diamagnets) respond differently to magnetic fields, leading to magnetization.
2. Key Points
| Concept | Symbol | Formula / Rule | What It Tells You |
|---|---|---|---|
| Magnetic field of a long straight wire | (B) | (\displaystyle B = \frac{\mu_0 I}{2\pi r}) | Field strength falls off with distance (r). |
| Biot–Savart Law | (d\mathbf{B}) | (\displaystyle d\mathbf{B}= \frac{\mu_0}{4\pi}\frac{I,d\mathbf{l}\times \hat{r}}{r^2}) | Gives field from any current element. |
| Ampère’s Law | (\oint \mathbf{B}\cdot d\mathbf{l}) | (\displaystyle \oint \mathbf{B}\cdot d\mathbf{l} = \mu_0 I_{\text{enc}}) | Useful for highly symmetric situations. |
| Magnetic field of a circular loop | (B_{\text{loop}}) | (\displaystyle B = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}) (at distance (x) from centre) | Field is strongest at the centre. |
| Force on a current element | (\mathbf{F}) | (\displaystyle d\mathbf{F}= I,d\mathbf{l}\times \mathbf{B}) | Determines how wires bend in a field. |
| Force between two parallel currents | (F/L) | (\displaystyle \frac{F}{L}= \frac{\mu_0 I_1 I_2}{2\pi d}) | Attraction or repulsion depending on current directions. |
| Torque on a current loop | (\tau) | (\displaystyle \tau = IAB\sin\theta) | (A) = area, (\theta) = angle between (\mathbf{B}) and normal. |
| Magnetic flux density | (B) | (B = \mu H) | (H) = magnetic field strength, (\mu) = permeability. |
| Magnetisation | (\mathbf{M}) | (\mathbf{B} = \mu_0(\mathbf{H} + \mathbf{M})) | Describes induced magnetic dipoles in materials. |
| Susceptibility | (\chi_m) | (\mathbf{M} = \chi_m \mathbf{H}) | Positive (\chi_m) → paramagnetism; negative → diamagnetism; large positive → ferromagnetism. |
| Magnetic dipole moment | (\mathbf{m}) | (\mathbf{m} = I\mathbf{A}) | Characterises the strength of a magnetic source. |
| Energy in a field | (U) | (\displaystyle U = -\mathbf{m}\cdot \mathbf{B}) | Minimum energy when (\mathbf{m}) aligns with (\mathbf{B}). |
Units & Constants
- ( \mu_0 = 4\pi \times 10^{-7}\ \text{T·m/A}) (permeability of free space).
- (B) in teslas (T), (H) in A/m, (I) in amperes (A), (A) in m², (d) in meters (m).
3. Worked Example
Problem:
A circular loop of radius (R = 0.10\ \text{m}) carries a current (I = 5\ \text{A}).
a) Find the magnetic field at the centre of the loop.
b) If the loop is placed in a uniform external field (B_{\text{ext}} = 0.02\ \text{T}) directed perpendicular to the plane of the loop (i.e., along the loop’s normal), calculate the torque on the loop.
a) Field at the centre
For a circular loop, the field at its centre is
[ B_{\text{center}} = \frac{\mu_0 I}{2R} ]
Plugging in the numbers:
[ \begin{aligned} B_{\text{center}} &= \frac{(4\pi \times 10^{-7}\ \text{T·m/A}) \times 5\ \text{A}}{2 \times 0.10