Magnetic Effects of Current & Magnetism Summary

1. Biot-Savart Law

The Biot-Savart law gives the magnetic field produced by a current-carrying element. The direction of the magnetic field element \( dB \) is perpendicular to both the current element \( Idl \) and the position vector \( \vec{r} \).

\[dB = \frac{\mu_0}{4\pi} \frac{Idl \sin \theta}{r^2}\]

Key Points:

The direction of \( \vec{dB} \) can be obtained by applying the right-hand rule to \( Id\vec{l} \times \vec{r} \)
\( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \) T·m/A)
\( \theta \) is the angle between the current element and the position vector

1.1 Magnetic Field Due to Different Current Systems

Current System	Magnetic Field Expression
Straight wire of finite length	\( B = \frac{\mu_0 I}{4 \pi R} (\sin \alpha_1 + \sin \alpha_2) \)
Infinitely long straight wire	\( B = \frac{\mu_0 I}{2 \pi R} \)
Arc of a circle	\( B = \frac{\mu_0 I \phi}{4 \pi R} \)
On the axis of a ring	\( B_{\parallel} = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}} \) \( B_{\perp} = 0 \)
At the center of a ring	\( B = \frac{\mu_0 I}{2R} \)
On the axis of a solenoid	\( B = \mu_0 n I = \mu_0 \frac{N}{l} I \)

1.2 Ampere's Circuital Law

The line integral of the magnetic field \( \vec{B} \) around any closed loop is equal to \( \mu_0 \) times the net current enclosed by the loop.

\[\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}\]

2. Magnetic Force

2.1 Force on a Moving Charge

A charged particle moving in a magnetic field experiences a force perpendicular to both its velocity and the magnetic field.

\[ \vec{F} = q(\vec{v} \times \vec{B}) \] \[ F = qvB \sin \theta \]

Where:

\( q \) = charge of the particle
\( \vec{v} \) = velocity of the particle
\( \vec{B} \) = magnetic field
\( \theta \) = angle between velocity and magnetic field vectors

2.2 Motion of a Charged Particle in a Uniform Magnetic Field

Circular Motion

When a charged particle moves perpendicular to a uniform magnetic field, it follows a circular path.

\[ qvB = \frac{mv^2}{r} \] \[ r = \frac{mv}{qB} \]

Where \( r \) is the radius of the circular path.

Time Period

The time period of revolution is independent of the velocity and radius.

\[ T = \frac{2\pi r}{v} = \frac{2\pi m}{qB} \]

2.3 Helical Motion

When a charged particle's velocity is not perpendicular to the magnetic field, it moves in a helical path.

\[ v = v_{\parallel} + v_{\perp} \] \[ \text{Radius of helix: } r = \frac{mv_{\perp}}{qB} \] \[ \text{Pitch: } p = v_{\parallel} T = \frac{2\pi mv_{\parallel}}{qB} \]

2.4 Lorentz Force

When a particle is subjected to both electric and magnetic fields, the total force is called the Lorentz force.

\[ \vec{F} = q(\vec{E} + \vec{v} \times \vec{B}) \]

2.5 Force on a Current-Carrying Conductor

A current-carrying conductor placed in a magnetic field experiences a force.

\[ d\vec{F} = I d\vec{l} \times \vec{B} \] \[ \vec{F} = \int (I d\vec{l} \times \vec{B}) \]

3. Magnetic Dipole

A small current-carrying loop acts like a magnetic dipole. The magnetic dipole moment is defined as the product of current and area enclosed by the loop.

\[ \vec{p}_m = I\vec{A} \] \[ \text{For N turns: } \vec{p}_m = NI\vec{A} \]

Key Points:

The direction of the magnetic moment coincides with the direction of the area vector
The area vector is perpendicular to the plane of the loop (right-hand rule)

3.1 Comparison: Electric vs Magnetic Dipole

Property	Electric Dipole	Magnetic Dipole
Field along axis	\( \vec{E}_{\parallel} = \frac{1}{2\pi\epsilon_0} \frac{\vec{p}_E}{x^3} \)	\( \vec{B}_{\parallel} = \frac{\mu_0 \vec{p}_m}{2\pi x^3} \)
Torque in external field	\( \vec{\tau} = \vec{p}_E \times \vec{E} \)	\( \vec{\tau} = \vec{p}_m \times \vec{B} \)
Potential energy	\( U = -\vec{p}_E \cdot \vec{E} \)	\( U = -\vec{p}_m \cdot \vec{B} \)

3.2 Field Due to a Bar Magnet

On Axial Line (End-on Position)

\[ B = \frac{\mu_0}{4\pi} \frac{2md}{(d^2 - l^2)^2} \] \[ \text{If } l^2 \ll d^2, \quad B = \frac{\mu_0}{4\pi} \frac{2m}{d^3} \]

On Equatorial Line (Broadside-on Position)

\[ B = \frac{\mu_0}{4\pi} \frac{m}{(d^2 + l^2)^{3/2}} \] \[ \text{If } l^2 \ll d^2, \quad B = \frac{\mu_0}{4\pi} \frac{m}{d^3} \]

3.3 Tangent Law and Tangent Galvanometer

When a magnet is suspended in two mutually perpendicular magnetic fields \( B \) and \( H \), it comes to rest at an angle \( \theta \) such that:

\[ B = H \tan \theta \]

Tangent Galvanometer

A tangent galvanometer measures electric current using the tangent law.

\[ B_H \tan \theta = \frac{\mu_0 i n}{2 r} \] \[ i = \frac{2 r B_H}{\mu_0 n} \tan \theta = K \tan \theta \]

Where \( K = \frac{2 r B_H}{\mu_0 n} \) is the galvanometer constant.

4. Magnetic Materials

4.1 Key Definitions

Magnetic Field (B): The total number of lines of force per unit area. Unit: Wb/m² or Tesla.

\[ \vec{B} = \frac{\mu}{4\pi} \frac{m}{r^2} \hat{r} \quad \text{Wb/m}^2 \] \[ \mu = \mu_0 \mu_r \]

Magnetic Field Strength (H): The magnetizing field, independent of the medium.

\[ \vec{H} = \frac{\vec{B}}{\mu} \quad \text{A/m} \]

Intensity of Magnetization (I): Magnetic moment per unit volume.

\[ I = \frac{\text{magnetic moment}}{\text{volume}} \]

Magnetic Susceptibility (χ): Ratio of intensity of magnetization to magnetizing field.

\[ \chi = \frac{I}{H} \]

Magnetic Permeability (μ): Ratio of magnetic induction to magnetizing field.

\[ \mu = \frac{B}{H} = \mu_0 \mu_r \]

4.2 Types of Magnetic Materials

Property	Diamagnetic	Paramagnetic	Ferromagnetic
Susceptibility (χ)	Small and negative	Small and positive	Large and positive
Relative permeability (μ_r)	< 1	> 1	>> 1
Behavior in non-uniform field	Repelled	Attracted	Strongly attracted
Field lines	Do not cross through	Cross through	Cross through
Examples	Cu, Zn, Bi, Ag, Au, Glass, NaCl	Al, Na, Sb, Pt, Mn, Cr	Fe, Ni, Co

4.3 Ferromagnetic Substances

Properties:

Acquire high degree of magnetization in the same sense as the applied field
Have permeability of the order of hundreds and thousands
Susceptibility is very large and positive
Attracted even by weak magnets
Above Curie temperature, they become paramagnetic

4.4 Hysteresis Loop

The hysteresis loop shows the relationship between magnetic field strength (H) and magnetic flux density (B) when a ferromagnetic material undergoes a complete cycle of magnetization.

Key Points:

The area enclosed by the hysteresis loop represents energy loss per unit volume per cycle
Soft magnetic materials have narrow hysteresis loops (low energy loss)
Hard magnetic materials have wide hysteresis loops (high energy loss, good for permanent magnets)

5. Earth's Magnetism

5.1 Components of Earth's Magnetic Field

Earth's Magnetic Field Components:

Geographic meridian, Magnetic meridian, Angle of declination (δ), Angle of dip (φ)

\[ B_H = B \cos \phi \] \[ B_V = B \sin \phi \] \[ B_H^2 + B_V^2 = B^2 \]

5.2 Key Definitions

Angle of Declination (δ): The angle between the geographic meridian and the magnetic meridian.

Angle of Dip or Inclination (φ): The angle that the Earth's magnetic field makes with the horizontal.

Key Points:

At poles: φ = 90°
At equator: φ = 0°

Horizontal Component (H): The component of Earth's magnetic field in the horizontal direction.

\[ H = B_e \cos \delta \]

Vertical Component (V): The component of Earth's magnetic field in the vertical direction.

\[ V = B_e \sin \delta \]