LEC 3: Magnetic Field: Properties, Forces, and Laws (Electrodynamics & Relativity – Lecture 25)

Magnetic Field: Properties, Forces, and Laws

Overview of Electromagnetic Concepts

This topic covers the fundamental properties of magnetic fields, their origins, and the laws governing their behavior. It is essential for understanding the relationship between electricity and magnetism, as well as the forces exerted on moving charges and currents.

• Early Electromagnetic Discoveries: Experiments by Coulomb, Ørsted, and Ampère revealed links between electricity and magnetism.

• Field Concept: Faraday introduced the concept of electric and magnetic fields, describing how changing one could produce effects in the other.

• Unification by Maxwell: Maxwell's equations unify the behavior of electric and magnetic fields and predict electromagnetic wave propagation.

Magnets

Magnets are objects that produce magnetic fields and have distinct poles. Their properties are foundational to understanding magnetic phenomena.

• Magnetic Poles: Magnets have two types of poles: north and south.

• Earth's Magnetic Poles: North magnetic poles are attracted toward Earth's geographic North Pole.

• Interaction of Poles: Like poles repel, unlike poles attract.

• No Magnetic Monopoles: Splitting a magnet results in two smaller magnets, each with both a north and south pole.

• Magnetic Field Generation: Magnets generate a magnetic field vector $\vec{B}$.

Magnetic Field Lines

Magnetic fields can be visualized using field lines, which illustrate the direction and strength of the field.

• Properties of Field Lines:

  • Field is tangent to the magnetic field line (points from N→S outside the dipole).

  • Field strength is proportional to line density.

  • Field lines never cross.

  • Field lines form continuous, closed loops.

• No Magnetic Monopoles: Field lines have no start or end points; $\nabla \cdot \vec{B} = 0$ (over the whole domain).

• Superposition Principle: Magnetic fields from multiple currents add vectorially.

Magnetic Force on Moving Charges

Charged particles moving in a magnetic field experience a force perpendicular to both their velocity and the magnetic field.

• Force Equation: $\vec{F} = q\vec{v} \times \vec{B}$

• Direction: Determined by the right-hand rule.

• Work Done: Magnetic force does no work; it only changes the direction of the velocity.

• Trajectory: For constant $\vec{B}$, the path is circular with radius $r = \frac{mv}{|q|B}$.

• Helical Path: If velocity has a component along $\vec{B}$, the trajectory is helical.

Force on a Current-Carrying Wire Segment

Current-carrying wires in a magnetic field experience a force due to the movement of charges.

• Current: Ordered movement of charge.

• Force on Wire: For a straight wire, $\vec{F} = I\vec{l} \times \vec{B}$, where $I$ is current and $\vec{l}$ is the length vector.

• Derivation: The total force is the sum of forces on individual charge carriers.

Magnetic Field Lines Around a Current-Carrying Wire

Electric currents produce magnetic fields, which form concentric circles around the wire.

• Right-Hand Rule: Thumb points in direction of current, fingers curl in direction of magnetic field lines.

• Field Lines: Magnetic field lines encircle the wire.

Biot-Savart Law

The Biot-Savart law quantitatively relates magnetic fields to the currents that produce them.

• Law Statement: The magnetic field $d\vec{B}$ at a point due to a current element is: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \vec{r}}{r^3}$

• Parameters:

  • $\mu_0$ = permeability of free space

  • $I$ = current

  • $d\vec{l}$ = current element vector

  • $\vec{r}$ = vector from current element to point

  • $r$ = distance from element to point

• Dependence: Magnitude depends on current strength, distance, and angle.

Example Calculation: Magnetic Field at Center of Circular Loop

Calculating the magnetic field at the center of a circular loop using the Biot-Savart law.

• Setup: Circular loop of radius $R$ carrying current $I$.

• Biot-Savart Law: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \vec{r}}{r^3}$

• Result: $B = \frac{\mu_0 I}{2R}$

Statement of Ampere’s Law (Integral Form)

Ampere’s law relates the integrated magnetic field around a closed loop to the current passing through the loop.

• Integral Form: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enclosed}}$

• Meaning: The closed loop integral of $\vec{B}$ equals the permeability times the enclosed current.

Ampere’s Law Applied to Long Straight Wire

Using Ampere’s law to find the magnetic field around a long, straight wire.

• Result: $B = \frac{\mu_0 I}{2\pi r}$, where $r$ is the distance from the wire.

• Field Direction: Circular around the wire (right-hand rule).

Ampere’s Law: Solenoids

A solenoid is a coil of wire that generates a uniform magnetic field inside when current flows through it.

• Field Inside Solenoid: $B = \mu_0 n I$, where $n$ is the number of turns per unit length.

• Field Direction: Along the axis of the solenoid.

Ampere’s Law Applied to Toroids

A toroid is a donut-shaped coil. Ampere’s law can be used to find the magnetic field inside the toroid.

• Field Inside Toroid: $B = \frac{\mu_0 N I}{2\pi r}$, where $N$ is the number of windings and $r$ is the radius of the circular path inside the toroid.

• Field Outside Toroid: $B = 0$ (no enclosed current).

Summary Table: Key Laws and Equations

Law/Equation	Formula (LaTeX)	Description
Magnetic Force on Charge	$\vec{F} = q\vec{v} \times \vec{B}$	Force on moving charge in magnetic field
Force on Wire	$\vec{F} = I\vec{l} \times \vec{B}$	Force on current-carrying wire in magnetic field
Biot-Savart Law	$d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \vec{r}}{r^3}$	Magnetic field from current element
Field at Center of Loop	$B = \frac{\mu_0 I}{2R}$	Magnetic field at center of circular loop
Ampere’s Law (Integral)	$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enclosed}}$	Magnetic field around closed loop
Field Around Wire	$B = \frac{\mu_0 I}{2\pi r}$	Field at distance $r$ from long straight wire
Field Inside Solenoid	$B = \mu_0 n I$	Uniform field inside solenoid
Field Inside Toroid	$B = \frac{\mu_0 N I}{2\pi r}$	Field inside toroid

Summary

• The magnetic field vector $\vec{B}$ describes the magnetic influence at a point in space and is essential for calculating forces on moving charges and currents.

• Magnetic field lines form continuous loops, never intersect, and their density indicates field strength.

• The Biot-Savart law relates magnetic fields to currents, with the field at a point depending on current, distance, and angle.

• Ampere’s law states that the magnetic field around a closed loop is proportional to the enclosed current: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enclosed}}$.