Magnetic Field due to Current — Revision Notes

⚡ 30-Second Revision

Ørsted's Discovery: — Current creates magnetic field.
Right-Hand Thumb Rule: — Thumb = current, fingers = B-field direction.
Biot-Savart Law: — $dvec{B} = \frac{\mu_0}{4pi} \frac{I dvec{l} \times \vec{r}}{r^3}$ (differential, universal).
Ampere's Circuital Law: — $\oint \vec{B} \cdot dvec{l} = \mu_0 I_{enclosed}$ (integral, for symmetry).
Straight Wire (infinite): — $B = \frac{\mu_0 I}{2\pi r}$
Circular Loop (center): — $B = \frac{\mu_0 I}{2R}$ (for N turns: $B = \frac{\mu_0 N I}{2R}$)
Circular Loop (axis): — $B = \frac{\mu_0 I R^2}{2(R^2+x^2)^{3/2}}$
Solenoid (inside): — $B = \mu_0 n I$ ($n$ = turns/length)
Toroid (inside): — $B = \frac{\mu_0 N I}{2\pi r}$ ($N$ = total turns, $r$ = mean radius)
$\mu_0$ (Permeability of Free Space): — $4\pi \times 10^{-7},\text{T}\cdot\text{m/A}$

2-Minute Revision

The core idea is that moving charges, or electric currents, generate magnetic fields. This was first observed by Ørsted. The direction of these fields is crucial and is determined by the Right-Hand Thumb Rule: for a straight wire, thumb points in current direction, curled fingers show field direction; for a loop, fingers curl with current, thumb shows field direction inside.

Quantitatively, the Biot-Savart Law ($dvec{B} = \frac{\mu_0}{4pi} \frac{I dvec{l} \times \vec{r}}{r^3}$) gives the field from a small current element and is universally applicable. For highly symmetric current distributions, Ampere's Circuital Law ($\oint \vec{B} \cdot dvec{l} = \mu_0 I_{enclosed}$) provides a simpler integral approach.

Key formulas to remember are for an infinite straight wire ($B = \frac{\mu_0 I}{2\pi r}$), a circular loop at its center ($B = \frac{\mu_0 I}{2R}$), inside a solenoid ($B = \mu_0 n I$), and inside a toroid ($B = \frac{\mu_0 N I}{2\pi r}$).

Remember that magnetic fields are vector quantities, so for multiple sources, vector addition is required. Always pay attention to units and the value of $\mu_0$.

5-Minute Revision

Revisiting 'Magnetic Field due to Current' means solidifying your understanding of how electricity creates magnetism. Start with Ørsted's discovery: current in a wire produces a magnetic field. The Right-Hand Thumb Rule is your best friend for direction: for a straight wire, thumb with current, fingers curl in B-field direction; for a loop, fingers with current, thumb points along axial B-field.

Two fundamental laws govern calculations:

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Biot-Savart Law: — $dvec{B} = \frac{\mu_0}{4pi} \frac{I dvec{l} \times \vec{r}}{r^3}$. This is a differential law, meaning it calculates the field from a tiny current segment. It's universal but often requires complex integration. Remember the cross product for direction.
2
Ampere's Circuital Law: — $\oint \vec{B} \cdot dvec{l} = \mu_0 I_{enclosed}$. This is an integral law, incredibly useful for symmetric current distributions (like infinite wires, solenoids, toroids) because it simplifies the integral. Choose your Amperian loop wisely!

Key Formulas to Master:

Infinite Straight Wire: — $B = \frac{\mu_0 I}{2\pi r}$. Example: A $10\,\text{A}$ current in a long wire. At $1\,\text{cm}$ ($0.01\,\text{m}$) away, $B = \frac{4\pi \times 10^{-7} \times 10}{2\pi \times 0.01} = 2 \times 10^{-4},\text{T}$.
Circular Loop (at center): — $B = \frac{\mu_0 I}{2R}$. For $N$ turns, $B = \frac{\mu_0 N I}{2R}$. Example: A 5-turn coil of $5\,\text{cm}$ radius with $2\,\text{A}$ current. $B = \frac{4\pi \times 10^{-7} \times 5 \times 2}{2 \times 0.05} = 4\pi \times 10^{-5},\text{T}$.
Circular Loop (on axis): — $B = \frac{\mu_0 I R^2}{2(R^2+x^2)^{3/2}}$. This shows B decreases with distance $x$.
Solenoid (inside): — $B = \mu_0 n I$, where $n$ is turns per unit length. Example: Solenoid with $2000\,\text{turns/m}$ and $1\,\text{A}$ current. $B = 4\pi \times 10^{-7} \times 2000 \times 1 = 8\pi \times 10^{-4},\text{T}$.
Toroid (inside): — $B = \frac{\mu_0 N I}{2\pi r}$, where $N$ is total turns and $r$ is mean radius.

Remember that magnetic fields are vector quantities. If you have multiple current sources, you must find the field from each and then add them vectorially, considering both magnitude and direction. Always check units and use $\mu_0 = 4\pi \times 10^{-7},\text{T}\cdot\text{m/A}$. Common mistakes include confusing direction, incorrect formula application, or unit errors. Practice problems involving superposition and varying geometries to solidify your understanding.

Prelims Revision Notes

1
Ørsted's Experiment: — Electric current produces magnetic field. This is the fundamental link.
2
Right-Hand Thumb Rule:

* Straight Wire: Thumb in current direction, curled fingers give magnetic field direction (concentric circles). * Circular Loop: Fingers curl in current direction, thumb gives magnetic field direction along the axis (inside the loop).

1
Biot-Savart Law:

* Differential form: $dvec{B} = \frac{\mu_0}{4pi} \frac{I dvec{l} \times \vec{r}}{r^3}$. * Magnitude: $dB = \frac{\mu_0}{4pi} \frac{I dl \sin\theta}{r^2}$. * $\mu_0 = 4\pi \times 10^{-7},\text{T}\cdot\text{m/A}$ (permeability of free space). * Used for any current distribution, especially for infinitesimal elements.

1
Magnetic Field due to a Long Straight Wire:

* Infinite wire: $B = \frac{\mu_0 I}{2\pi r}$. Direction: concentric circles, by RHTR. * Finite wire: $B = \frac{\mu_0 I}{4\pi r} (\sin\theta_1 + \sin\theta_2)$.

1
Magnetic Field due to a Circular Loop:

* At center: $B = \frac{\mu_0 I}{2R}$. For $N$ turns: $B = \frac{\mu_0 N I}{2R}$. * On axis (at distance $x$ from center): $B = \frac{\mu_0 I R^2}{2(R^2+x^2)^{3/2}}$.

1
Ampere's Circuital Law:

* Integral form: $\oint \vec{B} \cdot dvec{l} = \mu_0 I_{enclosed}$. Used for highly symmetric current distributions. $I_{enclosed}$ is the net current passing through the Amperian loop, direction by RHTR for the loop.

1
Magnetic Field Inside a Solenoid:

* Long solenoid: $B = \mu_0 n I$, where $n = N/L$ (turns per unit length). * Field is uniform and axial inside, negligible outside.

1
Magnetic Field Inside a Toroid:

* $B = \frac{\mu_0 N I}{2\pi r}$, where $N$ is total turns, $r$ is mean radius. * Field is confined within the toroid, zero outside.

1
Superposition Principle: — For multiple current sources, the net magnetic field at a point is the vector sum of the fields produced by individual sources.
2
Key Points: — Magnetic field is a vector. Field lines are closed loops. Pay attention to units (cm to m, etc.) and powers of 10 in calculations.

Vyyuha Quick Recall

For Biot-Savart, remember: 'B-I-L-S-I-N-R-Squared'. B is proportional to I, L (dl), Sin(theta), and inversely to R-squared. (Ignoring constants and vector nature for quick recall of dependencies). For Ampere's Law, think 'Amps Enclose Current'. The integral of B around a loop equals mu-naught times the enclosed current.