Conductivity, Resistance, and Magnetic Fields: Study Notes ch 29

Conductivity and Resistance

Conductivity

Conductivity is a measure of a material's ability to conduct electric current. It is the reciprocal of resistivity and is an intrinsic property of the material, independent of its geometry.

• Definition: Conductivity ($\sigma$) is given by $\sigma = \frac{n e^2 \tau}{m}$, where $n$ is the number density of charge carriers, $e$ is the elementary charge, $\tau$ is the mean time between collisions, and $m$ is the mass of the charge carrier.

• Example: Copper has a higher conductivity than nichrome because its electrons have a longer mean time between collisions, resulting in fewer collisions and less energy loss.

• Application: Copper is used in household wiring for its high conductivity, while nichrome is used in toasters for its lower conductivity (more energy loss as heat).

Superconductivity

Superconductivity is a phenomenon where certain materials exhibit zero electrical resistance below a critical temperature. This allows current to flow without energy loss.

• Key Point: Magnetic fields do not penetrate superconductors (Meissner effect), causing effects such as magnetic levitation.

• Example: Superconductors can cause magnets to float above them due to the expulsion of magnetic fields.

Resistance and Ohm's Law

Resistance quantifies how much a material opposes the flow of electric current. Ohm's Law relates current, voltage, and resistance in ohmic materials.

• Ohm's Law: $I = \frac{\Delta V}{R}$, where $I$ is current, $\Delta V$ is potential difference, and $R$ is resistance.

• Resistance of a wire: $R = \rho \frac{L}{A}$, where $\rho$ is resistivity, $L$ is length, and $A$ is cross-sectional area.

• Example: Halving the voltage across an ohmic wire halves the current, since resistance remains constant.

Increasing Current in a Wire

The current through a wire can be increased by adjusting its physical properties or the material's resistivity.

• Increase the radius of the wire by a factor of 2 (increases area by 4, decreases resistance by 4, increases current by 4).

• Decrease the length of the wire by a factor of 4 (decreases resistance by 4, increases current by 4).

• Use a material with a resistivity a quarter that of the original wire (decreases resistance by 4, increases current by 4).

• Use a material with a resistivity four times that of the original wire (increases resistance by 4, decreases current by 4).

Magnetism and Magnetic Fields

Magnetic Dipoles and Poles

Magnets are magnetic dipoles, possessing a north and a south pole. Like poles repel, and opposite poles attract.

• Earth's Magnetism: The geographic north pole is near the Earth's magnetic south pole, as it attracts the north pole of a magnet.

• Breaking a Magnet: Each piece forms a new dipole with its own north and south poles.

Magnetic Field Lines

Magnetic fields are visualized using field lines, which indicate the direction and strength of the field.

• Field Line Properties: Tangent to the field at any point; closer lines indicate stronger fields.

• Compass Alignment: A compass needle aligns with the local magnetic field direction.

Magnetic Field from Currents

Electric currents produce magnetic fields. The direction of the field around a straight wire can be determined by the right-hand rule.

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

• Superposition Principle: The net magnetic field is the vector sum of fields from all sources: $\vec{B}_{\text{total}} = \vec{B}_1 + \vec{B}_2 + \ldots$

Magnetic Field from Moving Charges

A moving charge creates a magnetic field, described by the Biot-Savart law.

• Magnitude: $B_{\text{point charge}} = \frac{\mu_0 q v \sin \theta}{4 \pi r^2}$

• Direction: Determined by the right-hand rule for cross products ($\vec{v} \times \vec{r}$).

• Unit: Tesla (T), where $1\,\text{T} = 1\,\text{N}/(\text{A} \cdot \text{m})$

Magnetic Field of a Current-Carrying Wire

The magnetic field due to a current segment or a long straight wire is given by:

• $\vec{B}_{\text{current segment}} = \frac{\mu_0 I \Delta \vec{s} \times \hat{r}}{4 \pi r^2}$

• $B_{\text{wire}} = \frac{\mu_0 I}{2 \pi r}$ for a long straight wire

• $B_{\text{coil center}} = \frac{\mu_0 N I}{2 R}$ for a coil of radius $R$ with $N$ turns

Magnetic Field from Loops and Solenoids

Current loops and solenoids generate magnetic fields similar to those of bar magnets.

• Magnetic Dipole Moment: $m = IA$, where $I$ is current and $A$ is area.

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

• Application: MRI machines and doorbells use solenoids to generate strong, uniform magnetic fields.

Ampère's Law

Ampère's Law relates the magnetic field around a closed loop to the current passing through the loop:

• $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{\text{through}}$

• If $\vec{B}$ is tangent to the curve and constant in magnitude: $\oint \vec{B} \cdot d\vec{s} = B l$

Magnetic Force on Moving Charges

A charged particle moving in a magnetic field experiences a force perpendicular to both its velocity and the field.

• Force: $F_{\text{on}} = |q| v B \sin \theta$

• Direction: Given by the right-hand rule for $q > 0$, left-hand for $q < 0$.

• Circular Motion: If velocity is perpendicular to $\vec{B}$, the particle moves in a circle of radius $r_{\text{cyc}} = \frac{mv}{|q|B}$

Magnetic Force on Current-Carrying Wires

A wire carrying current in a magnetic field experiences a force given by:

• $\vec{F} = I \vec{L} \times \vec{B}$

• Direction is determined by the right-hand rule.

Magnetic Dipoles and Torques

A current loop in a magnetic field experiences a torque that tends to align the dipole moment with the field.

• Torque: $\vec{\tau} = \vec{m} \times \vec{B}$

• Potential Energy: $U = -\vec{m} \cdot \vec{B}$

Magnetic Properties of Matter

Materials respond differently to magnetic fields, classified as diamagnetic, paramagnetic, or ferromagnetic based on their atomic structure and response to external fields.

• Diamagnetic: Weakly repelled by magnetic fields.

• Paramagnetic: Weakly attracted by magnetic fields.

• Ferromagnetic: Strongly attracted and can retain magnetization (e.g., iron).

Summary Table: Key Equations and Concepts

Concept	Equation	Description
Ohm's Law	$I = \frac{\Delta V}{R}$	Current through a resistor
Resistance of a wire	$R = \rho \frac{L}{A}$	Depends on resistivity, length, and area
Conductivity	$\sigma = \frac{1}{\rho}$	Inverse of resistivity
Magnetic field (long straight wire)	$B = \frac{\mu_0 I}{2 \pi r}$	Field at distance $r$ from wire
Magnetic field (solenoid)	$B = \mu_0 n I$	Uniform field inside solenoid
Magnetic force on charge	$F = |q| v B \sin \theta$	Force on moving charge in $\vec{B}$
Magnetic force on wire	$\vec{F} = I \vec{L} \times \vec{B}$	Force on current-carrying wire
Ampère's Law	$\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{\text{through}}$	Relates field around closed loop to current

Additional info: Some context and explanations have been expanded for clarity and completeness, including the summary table and applications.