Maharashtra HSC · Class XII · Physics

Chapter 10: Magnetic Fields due to Electric Current

Full revision notes for Chapter 10 - Magnetic Fields due to Electric Current.

10.1

Biot-Savart Law

d\vec{B} = \frac{\mu_0}{4\pi}\,\frac{I\,d\vec{l}\times\vec{r}}{r^3}

dB = \frac{\mu_0}{4\pi}\,\frac{I\,dl\,\sin\theta}{r^2}

Field due to a current element. $I$ current, $dl$ element length, $r$ distance, $\theta$ angle between the element and the radius vector and $\mu_0 = 4\pi\times 10^{-7}$ N/A² the permeability of free space.

Long straight wire (infinite)

B = \frac{\mu_0 I}{2\pi r}

At perpendicular distance $r$ from a very long (effectively infinite) wire. Field lines are concentric circles; direction from the right-hand rule.

Circular loop (radius R)

B_{center} = \frac{\mu_0 I}{2R} \qquad B_{N,center} = \frac{\mu_0 N I}{2R}

B_{axis}(x) = \frac{\mu_0 I R^2}{2\,(R^2 + x^2)^{3/2}}

Center field for a single loop and for $N$ turns; on the axis at distance $x$ from the center. For a circular arc subtending angle $\theta$ , $B = \mu_0 I\theta/(4\pi R)$ .

10.2

Ampère's Circuital Law and Solenoids

Ampère's law

\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{enclosed}

The line integral of $\vec{B}$ around a closed path equals $\mu_0$ times the net current enclosed. Useful for highly symmetric geometries.

Long solenoid

B = \mu_0 n I

$n = N/L$ turns per unit length. Uniform, strong field inside; nearly zero outside (ideal, long solenoid).

Toroid

B = \frac{\mu_0 N I}{2\pi r}

Inside the toroid at radius $r$ , for $N$ turns; field is confined to the interior (ideal toroid).

10.3

Forces on Charges and Currents

Force on a moving charge

\vec{F} = q\,\vec{v}\times\vec{B} \qquad F = qvB\sin\theta

Magnetic force on a charge; perpendicular to both velocity and field, so it does no work. Lorentz force with an electric field: $\vec{F} = q(\vec{E} + \vec{v}\times\vec{B})$ . Field units: tesla (T).

Force on a current-carrying wire

F = B I l \sin\theta

Force on a straight wire of length $l$ carrying current $I$ at angle $\theta$ to the field.

Force between two parallel wires

\frac{F}{l} = \frac{\mu_0 I_1 I_2}{2\pi d}

Force per unit length between parallel wires separated by $d$ . Parallel currents attract, antiparallel currents repel. This defines the ampere.

10.4

Moving-Coil Galvanometer and Conversion

Galvanometer

\theta \propto I \qquad \theta = \frac{N B A}{C}I

Deflection proportional to current. $N$ turns, $B$ field, $A$ coil area and $C$ the torsional constant. Current sensitivity $\theta/I = NBA/C$ ; voltage sensitivity $\theta/V = NBA/(C\,G)$ with galvanometer resistance $G$ .

To an ammeter (low shunt in parallel)

S = \frac{I_g G}{I - I_g}

Shunt $S$ carries the excess current; $I_g$ is the full-scale galvanometer current and $I$ the new range. Ammeters are low-resistance, connected in series.

To a voltmeter (high multiplier in series)

R = \frac{V}{I_g} - G

Series resistance $R$ extends the range to $V$ . Voltmeters are high-resistance, connected in parallel.