This topic covers magnetic fields due to currents and the behavior of magnetic materials.
Overview
graph TD
A[Magnetism] --> B[Magnetic Field]
A --> C[Forces]
A --> D[Magnetic Materials]
B --> B1[Biot-Savart Law]
B --> B2[Ampere's Law]
C --> C1[Moving Charge]
C --> C2[Current Conductor]
D --> D1[Diamagnetic]
D --> D2[Paramagnetic]
D --> D3[Ferromagnetic]Biot-Savart Law
Magnetic field due to current element:
$$\boxed{d\vec{B} = \frac{\mu_0}{4\pi} \frac{Id\vec{l} \times \hat{r}}{r^2}}$$where $\mu_0 = 4\pi \times 10^{-7}$ T⋅m/A
Magnetic Field Due to Straight Wire
Infinite wire:
$$\boxed{B = \frac{\mu_0 I}{2\pi r}}$$Finite wire (angles α and β from ends):
$$B = \frac{\mu_0 I}{4\pi r}(\sin\alpha + \sin\beta)$$Magnetic Field at Center of Circular Loop
$$\boxed{B = \frac{\mu_0 NI}{2R}}$$where N = number of turns
Magnetic Field on Axis of Circular Loop
$$B = \frac{\mu_0 NIR^2}{2(R^2 + x^2)^{3/2}}$$Ampere’s Circuital Law
$$\boxed{\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enclosed}}$$Applications
Inside solenoid:
$$\boxed{B = \mu_0 nI}$$where n = turns per unit length
Toroid:
$$B = \frac{\mu_0 NI}{2\pi r}$$Force on Moving Charge
$$\boxed{\vec{F} = q(\vec{v} \times \vec{B})}$$Magnitude: $F = qvB\sin\theta$
Motion of Charged Particle
In uniform B field:
If $\vec{v} \perp \vec{B}$: Circular motion
- Radius: $r = \frac{mv}{qB}$
- Period: $T = \frac{2\pi m}{qB}$
If $\vec{v}$ has component along $\vec{B}$: Helical motion
Lorentz Force
$$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$$Force on Current-Carrying Conductor
$$\boxed{\vec{F} = I(\vec{L} \times \vec{B})}$$Magnitude: $F = BIL\sin\theta$
Force Between Parallel Wires
$$\boxed{\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}}$$- Same direction currents: Attractive
- Opposite direction currents: Repulsive
Definition of Ampere: Current producing 2×10⁻⁷ N/m between infinite parallel wires 1 m apart.
Torque on Current Loop
$$\boxed{\vec{\tau} = \vec{M} \times \vec{B} = NIA\vec{n} \times \vec{B}}$$where $M = NIA$ = magnetic dipole moment
Magnitude: $\tau = NIAB\sin\theta$
Moving Coil Galvanometer
Deflection:
$$\theta = \frac{NIAB}{k}$$Current sensitivity:
$$S_I = \frac{\theta}{I} = \frac{NAB}{k}$$Conversion
To Ammeter: Add shunt $S = \frac{I_g G}{I - I_g}$ in parallel
To Voltmeter: Add resistance $R = \frac{V}{I_g} - G$ in series
Magnetic Dipole
Bar Magnet
Axial field (on axis):
$$B = \frac{\mu_0 2M}{4\pi r^3}$$Equatorial field (perpendicular bisector):
$$B = \frac{\mu_0 M}{4\pi r^3}$$Torque on Bar Magnet in Field
$$\tau = MB\sin\theta$$Potential Energy
$$U = -\vec{M} \cdot \vec{B} = -MB\cos\theta$$Magnetic Materials
Classification
| Property | Diamagnetic | Paramagnetic | Ferromagnetic |
|---|---|---|---|
| χ (susceptibility) | Small, negative | Small, positive | Large, positive |
| μᵣ (relative permeability) | < 1 | > 1 | » 1 |
| Effect of B | Weakly repelled | Weakly attracted | Strongly attracted |
| Examples | Bi, Cu, Ag, H₂O | Al, Na, O₂ | Fe, Co, Ni |
Curie Law
For paramagnetic materials:
$$\chi \propto \frac{1}{T}$$Hysteresis
- Retentivity: B when H = 0
- Coercivity: H needed to make B = 0
Practice Problems
Find the magnetic field at the center of a circular coil of 10 turns, radius 0.1 m, carrying 2 A.
Two long parallel wires 10 cm apart carry currents 5 A and 10 A in same direction. Find force per meter.
An electron moves in a circle of radius 2 cm in a magnetic field of 0.01 T. Find its velocity.
A rectangular loop (20 cm × 10 cm) carries 5 A in a field of 0.1 T. Find maximum torque.