Magnetic Effects of Current and Magnetism — Revision Notes

⚡ 30-Second Revision

Biot-Savart Law: — $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$
Field (Straight Wire): — $B = \frac{\mu_0 I}{2\pi r}$
Field (Circular Loop Center): — $B = \frac{\mu_0 N I}{2R}$
Field (Solenoid): — $B = \mu_0 n I$
Lorentz Force (Charge): — $\vec{F} = q(\vec{v} \times \vec{B})$ (no work done)
Lorentz Force (Conductor): — $\vec{F} = I(\vec{L} \times \vec{B})$
Force (Parallel Wires): — $\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$ (same direction: attract, opposite: repel)
Magnetic Dipole Moment: — $\vec{M} = N I \vec{A}$
Torque on Loop: — $\vec{\tau} = \vec{M} \times \vec{B}$
Radius of Circular Path (Charge): — $r = \frac{mv}{qB}$
Magnetic Materials: — Diamagnetic (repelled, $\chi_m < 0$), Paramagnetic (weakly attracted, $\chi_m > 0$, Curie's Law), Ferromagnetic (strongly attracted, domains, Curie Temp.)
Galvanometer to Ammeter: — Shunt ($R_s = \frac{I_g R_g}{I - I_g}$) in parallel.
Galvanometer to Voltmeter: — Series resistance ($R_{series} = \frac{V}{I_g} - R_g$) in series.

2-Minute Revision

Magnetic effects of current describe how moving charges create magnetic fields. The Biot-Savart Law quantifies this for current elements, leading to formulas for magnetic fields around straight wires ($B = \frac{\mu_0 I}{2\pi r}$), circular loops ($B = \frac{\mu_0 N I}{2R}$ at center), and solenoids ($B = \mu_0 n I$).

Ampere's Circuital Law provides an alternative for symmetric cases. A charged particle moving in a magnetic field experiences the Lorentz force ($\vec{F} = q(\vec{v} \times \vec{B})$), which is always perpendicular to velocity, doing no work and thus not changing the particle's speed or kinetic energy.

This force causes circular or helical motion. Current-carrying conductors also experience force ($\vec{F} = I(\vec{L} \times \vec{B})$), leading to attraction/repulsion between parallel wires. Materials are classified as diamagnetic (repelled), paramagnetic (weakly attracted, temperature-dependent), or ferromagnetic (strongly attracted, domains, Curie temperature).

Earth's magnetic field is described by declination and dip. Galvanometers, based on torque on current loops, can be converted to ammeters (shunt in parallel) or voltmeters (high resistance in series).

Remember right-hand rules for directions.

5-Minute Revision

This chapter is about the fundamental connection between electricity and magnetism. Moving charges (electric current) produce magnetic fields, a discovery by Ørsted. The direction of these fields is found using the Right-Hand Thumb Rule.

Quantitatively, the Biot-Savart Law ($d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$) allows calculating the magnetic field from any current distribution. Key applications include the field due to a long straight wire ($B = \frac{\mu_0 I}{2\pi r}$), at the center of a circular loop ($B = \frac{\mu_0 N I}{2R}$), and inside a solenoid ($B = \mu_0 n I$).

For highly symmetric cases, Ampere's Circuital Law ($\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$) is more convenient.

When a charged particle moves in a magnetic field, it experiences the Lorentz force, $\vec{F} = q(\vec{v} \times \vec{B})$. This force is always perpendicular to both velocity and the magnetic field, meaning it does no work and thus cannot change the particle's speed or kinetic energy, only its direction.

This leads to circular motion ($r = \frac{mv}{qB}$) if velocity is perpendicular to the field, or helical motion if there's a parallel component. A current-carrying conductor of length $L$ in a magnetic field $B$ experiences a force $\vec{F} = I(\vec{L} \times \vec{B})$.

This explains the force between two parallel current-carrying wires: they attract if currents are in the same direction and repel if opposite, with force per unit length $\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$.

Materials respond differently to magnetic fields: diamagnetic (weakly repelled, $\chi_m < 0$), paramagnetic (weakly attracted, $\chi_m > 0$, follows Curie's Law), and ferromagnetic (strongly attracted, exhibit domains, lose ferromagnetism above Curie temperature).

Earth's magnetism is characterized by declination and dip. The moving coil galvanometer works on the principle of torque on a current loop ($\vec{\tau} = \vec{M} \times \vec{B}$) in a magnetic field, and can be converted into an ammeter (low shunt resistance in parallel) or a voltmeter (high series resistance).

Worked Example: A proton moves with velocity $3 \times 10^5\,\text{m/s}$ in a region where a uniform magnetic field of $0.5\,\text{T}$ is directed perpendicular to its velocity. Find the magnetic force on the proton. ($q_p = 1.6 \times 10^{-19}\,\text{C}$) Solution: $F = qvB\sin\theta$. Here $\theta = 90^\circ$, so $\sin\theta = 1$. $F = (1.6 \times 10^{-19}\,\text{C}) \times (3 \times 10^5\,\text{m/s}) \times (0.5\,\text{T})$ $F = 2.4 \times 10^{-14}\,\text{N}$.

Prelims Revision Notes

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Magnetic Field Sources: — Moving charges (currents) produce magnetic fields (Ørsted's discovery).
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Right-Hand Thumb Rule: — For straight wire, thumb = current, curled fingers = B-field direction. For loop, curled fingers = current, thumb = B-field direction through loop.
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Biot-Savart Law: — $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$. $\mu_0 = 4\pi \times 10^{-7}\,\text{T}\,\text{m/A}$.
4
Magnetic Field Formulas:

* Long straight wire: $B = \frac{\mu_0 I}{2\pi r}$ * Circular loop (center): $B = \frac{\mu_0 N I}{2R}$ * Circular loop (axis, dist $x$): $B_x = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}$ * Solenoid (inside): $B = \mu_0 n I$ ($n$ = turns/unit length) * Toroid (inside): $B = \frac{\mu_0 N I}{2\pi r}$ ($r$ = mean radius)

1
Ampere's Circuital Law: — $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$. Useful for symmetric current distributions.
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Lorentz Force:

* On charge $q$: $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$. Magnetic force $\vec{F}_m = q(\vec{v} \times \vec{B})$. * Magnetic force is perpendicular to $\vec{v}$ and $\vec{B}$. It does NO WORK, so no change in speed or KE. * Motion of charge in uniform B-field (perpendicular $\vec{v}$): Circular path, radius $r = \frac{mv}{qB}$, frequency $f = \frac{qB}{2\pi m}$. * On current-carrying conductor: $\vec{F} = I(\vec{L} \times \vec{B})$.

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Force between Parallel Wires: — $\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$. Same direction currents ATTRACT, opposite direction currents REPEL.
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Magnetic Dipole Moment: — $\vec{M} = N I \vec{A}$. Direction by Right-Hand Rule (fingers current, thumb M).
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Torque on Current Loop: — $\vec{\tau} = \vec{M} \times \vec{B}$. Magnitude $\tau = MB\sin\theta$.
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Magnetic Materials:

* Diamagnetic: Weakly repelled. $\chi_m < 0$, $\mu_r < 1$. No permanent dipoles. Ex: Cu, H$_2$O. * Paramagnetic: Weakly attracted. $\chi_m > 0$, $\mu_r > 1$. Permanent dipoles, align with field. $\chi_m \propto 1/T$ (Curie's Law). Ex: Al, Na, O$_2$. * Ferromagnetic: Strongly attracted. $\chi_m \gg 0$, $\mu_r \gg 1$. Domains. Hysteresis. Lose ferromagnetism above Curie Temperature ($T_C$). Ex: Fe, Ni, Co.

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Earth's Magnetism:

* **Declination ($\alpha$):** Angle between geographic and magnetic meridians. * **Dip ($\delta$):** Angle of Earth's B-field with horizontal. $\tan\delta = B_V/B_H$.

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Moving Coil Galvanometer: — Principle: Torque on current loop. Current sensitivity $\propto NAB/k$.

* Ammeter Conversion: Low shunt resistance ($R_s = \frac{I_g R_g}{I - I_g}$) in PARALLEL. * Voltmeter Conversion: High resistance ($R_{series} = \frac{V}{I_g} - R_g$) in SERIES.

Vyyuha Quick Recall

For the types of magnetic materials and their properties: Don't Play Football.

Diamagnetic: Dislikes (repels) field, Doesn't have permanent dipoles, Decreases field. $\chi_m$ is Definitely negative.
Paramagnetic: Partially likes (attracts) field, Permanent dipoles (random), Positive $\chi_m$, Proportional to $1/T$ (Curie's Law).
Ferromagnetic: Fiercely likes (strongly attracts) field, Form domains, Formerly permanent magnets, Falls apart above Curie Temp.