Magnetic Field — Revision Notes

⚡ 30-Second Revision

Magnetic Field (B): — Vector field, measured in Tesla (T). Produced by moving charges/currents.
Sources: — Moving charges (currents), intrinsic magnetic moments.
Biot-Savart Law: — $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \vec{r}}{r^3}$
Ampere's Circuital Law: — $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$
Field - Straight Wire: — $B = \frac{\mu_0 I}{2\pi r}$
Field - Circular Loop (center): — $B = \frac{\mu_0 I}{2R}$
Field - Solenoid (inside): — $B = \mu_0 n I$
Lorentz Force: — $\vec{F} = q (\vec{v} \times \vec{B})$ (magnetic part)
Force on Current Wire: — $\vec{F} = I (\vec{L} \times \vec{B})$
Direction Rules: — Right-Hand Thumb Rule (straight wire), Right-Hand Curl Rule (loop/solenoid), Right-Hand Rule for cross products (Lorentz force).
Properties of Field Lines: — Closed loops, never intersect, density indicates strength, tangent gives direction.
Work by Magnetic Force: — Always zero ($W=0$).

2-Minute Revision

The magnetic field, denoted by $\vec{B}$ (Tesla), is a vector field generated by moving electric charges (currents) and the intrinsic magnetic moments of particles. Its direction is visualized by magnetic field lines, which always form continuous closed loops, never intersect, and whose density indicates field strength.

The fundamental laws governing magnetic fields are the Biot-Savart Law, which calculates the field due to an infinitesimal current element, and Ampere's Circuital Law, useful for highly symmetric current distributions.

Key formulas include $B = \frac{\mu_0 I}{2\pi r}$ for an infinite straight wire, $B = \frac{\mu_0 I}{2R}$ for a circular loop's center, and $B = \mu_0 n I$ for a long solenoid. A moving charge $q$ with velocity $\vec{v}$ in a magnetic field $\vec{B}$ experiences a Lorentz force $\vec{F} = q (\vec{v} \times \vec{B})$.

This force is always perpendicular to both $\vec{v}$ and $\vec{B}$, does no work, and thus cannot change the particle's speed, only its direction. Similarly, a current-carrying conductor of length $\vec{L}$ in a field $\vec{B}$ experiences a force $\vec{F} = I (\vec{L} \times \vec{B})$.

Mastering the right-hand rules for direction is crucial for solving problems.

5-Minute Revision

Magnetic fields are regions of influence around moving charges or magnets, described by the magnetic flux density $\vec{B}$ (in Tesla). They are fundamentally different from electric fields; magnetic fields only exert forces on *moving* charges. The two main laws for calculating magnetic fields are:

1
Biot-Savart Law: — For a current element $I d\vec{l}$, the field $d\vec{B}$ at a distance $\vec{r}$ is $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \vec{r}}{r^3}$. This is general but often requires integration.
2
Ampere's Circuital Law: — $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$. This is powerful for symmetric current distributions.

Key Formulas for Specific Geometries:

Infinite Straight Wire: — $B = \frac{\mu_0 I}{2\pi r}$ (field lines are concentric circles).
Center of Circular Loop: — $B = \frac{\mu_0 I}{2R}$.
Inside a Long Solenoid: — $B = \mu_0 n I$ (uniform, $n$ is turns per unit length).

Lorentz Force: The force on a charge $q$ moving with velocity $\vec{v}$ in a magnetic field $\vec{B}$ is $\vec{F} = q (\vec{v} \times \vec{B})$.

Direction: — Perpendicular to both $\vec{v}$ and $\vec{B}$ (use right-hand rule for cross product).
Work Done: — Zero. Magnetic force only changes direction, not speed or kinetic energy.
Circular Motion: — If $\vec{v} \perp \vec{B}$, the particle moves in a circle with radius $r = \frac{mv}{qB}$.

Force on Current-Carrying Conductor: For a wire of length $\vec{L}$ carrying current $I$ in a field $\vec{B}$, $\vec{F} = I (\vec{L} \times \vec{B})$.

Magnetic Field Lines: Always form closed loops (no monopoles), tangent gives field direction, density indicates strength, never intersect. Mastering the Right-Hand Rules for determining directions of fields and forces is absolutely critical for NEET problems.

Prelims Revision Notes

Magnetic fields are vector fields ($\vec{B}$, unit Tesla) produced by moving charges (currents) or intrinsic magnetic moments. The permeability of free space, $\mu_0 = 4\pi \times 10^{-7};\text{T}cdot\text{m/A}$, is a fundamental constant.

Sources and Laws:

1
Biot-Savart Law: — $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \vec{r}}{r^3}$. Used for any current distribution.
2
Ampere's Circuital Law: — $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$. Useful for high symmetry (e.g., infinite wire, solenoid, toroid).

Magnetic Field Magnitudes:

Infinite Straight Wire: — $B = \frac{\mu_0 I}{2\pi r}$ (r is perpendicular distance).
Center of Circular Loop (radius R, N turns): — $B = \frac{\mu_0 N I}{2R}$.
On Axis of Circular Loop (distance x from center): — $B = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}$.
Inside Long Solenoid (n turns/unit length): — $B = \mu_0 n I$ (uniform).
Inside Toroid (N total turns, average radius r): — $B = \frac{\mu_0 N I}{2\pi r}$.

Forces in Magnetic Fields:

1
Lorentz Force on Moving Charge: — $\vec{F} = q (\vec{v} \times \vec{B})$.

* Direction: Perpendicular to both $\vec{v}$ and $\vec{B}$. Use Right-Hand Rule for cross product. For negative charge, force is opposite. * Work Done: $W = 0$. Magnetic force does no work, thus cannot change kinetic energy or speed. * Circular Motion: If $\vec{v} \perp \vec{B}$, $qvB = \frac{mv^2}{r} \implies r = \frac{mv}{qB}$. Period $T = \frac{2\pi m}{qB}$, frequency $f = \frac{qB}{2\pi m}$.

1
Force on Current-Carrying Conductor: — $\vec{F} = I (\vec{L} \times \vec{B})$.

* Direction: Perpendicular to both $\vec{L}$ (current direction) and $\vec{B}$.

Magnetic Field Lines Properties:

Form continuous closed loops (no magnetic monopoles).
Tangent at any point gives the direction of $\vec{B}$.
Never intersect each other.
Closer lines indicate stronger field.

Direction Rules:

Right-Hand Thumb Rule: — For straight wire, thumb = current, curled fingers = field direction.
Right-Hand Curl Rule: — For loop/solenoid, curled fingers = current, thumb = field direction inside.
Right-Hand Rule for Cross Product: — For $\vec{A} \times \vec{B}$, point fingers along $\vec{A}$, curl towards $\vec{B}$, thumb gives direction of $\vec{A} \times \vec{B}$.

Remember to convert units (e.g., cm to m) and pay attention to vector directions and signs.

Vyyuha Quick Recall

For the direction of magnetic field around a current-carrying wire, remember 'Right-Hand Curl for Current': If your right thumb points in the direction of the current, your fingers curl in the direction of the magnetic field lines. For the force on a positive charge in a magnetic field, use 'V-B-F' (Velocity-B-Force) with Right Hand: Point your fingers in the direction of velocity (V), curl them towards the magnetic field (B), and your thumb will point in the direction of the force (F).