Introduction

This chapter explores the relationship between moving charges (electric currents) and magnetism. It begins with Oersted's discovery that electric currents create magnetic fields, leading to the understanding that moving charges are the source of magnetism. We will learn how magnetic fields exert forces on moving charges and current-carrying conductors, how currents generate magnetic fields (described by the Biot-Savart Law and Ampere's Circuital Law), and the principles behind devices like cyclotrons and galvanometers.

The convention for depicting magnetic fields and currents emerging from or entering a plane is introduced: a dot (¤) for fields/currents coming out, and a cross () for fields/currents going in.

Magnetic Force

Sources And Fields

Similar to how static charges create electric fields, moving charges and currents create magnetic fields ($\vec{B}$). Magnetic fields are vector fields that exist at every point in space and can vary with time (though this chapter focuses on static fields). They obey the principle of superposition, meaning the total magnetic field from multiple sources is the vector sum of the fields from each source.

Magnetic Field, Lorentz Force

When a charge $q$ moves with velocity $\vec{v}$ in the presence of both electric field $\vec{E}$ and magnetic field $\vec{B}$, it experiences the Lorentz force:

$\vec{F} = q [\vec{E} + (\vec{v} \times \vec{B})]$

The magnetic force component $\vec{F}_m = q(\vec{v} \times \vec{B})$ has the following properties:

• It depends on the charge $q$, velocity $\vec{v}$, and magnetic field $\vec{B}$.
• It is zero if the charge is not moving ($v=0$) or if the velocity is parallel or antiparallel to the magnetic field.
• It acts perpendicular to both $\vec{v}$ and $\vec{B}$, its direction given by the right-hand rule.
• It does no work on the charge as it is always perpendicular to the velocity.

The SI unit of magnetic field is the Tesla (T). A field of 1 Tesla exerts a force of 1 Newton on a charge of 1 Coulomb moving at 1 m/s perpendicular to the field.

Figure 4.2 illustrates the direction of the magnetic force.

Magnetic Force On A Current-Carrying Conductor

A conductor of length $\vec{l}$ carrying current $I$ in a uniform external magnetic field $\vec{B}$ experiences a force:

$\vec{F} = I (\vec{l} \times \vec{B})$

where $\vec{l}$ is a vector whose magnitude is the length of the conductor and whose direction is along the current flow. For arbitrary shapes, the force is the integral of $I d\vec{l} \times \vec{B}$ over the entire conductor.

Example 4.1: Calculates the magnetic field required to levitate a current-carrying wire against gravity.

Example 4.2: Demonstrates the direction of the Lorentz force on an electron and a proton moving in a magnetic field.

Motion In A Magnetic Field

When a charged particle moves in a uniform magnetic field, the magnetic force ($q\vec{v} \times \vec{B}$) does no work, as it is always perpendicular to the velocity. This means the kinetic energy and the speed of the particle remain constant, although its direction of motion can change.

• Velocity Perpendicular to Field: If the velocity $\vec{v}$ is perpendicular to $\vec{B}$, the magnetic force acts as a centripetal force, causing the particle to move in a circular path. The radius of the circle is $r = \frac{mv}{|q|B}$, and the frequency of revolution (cyclotron frequency) is $\nu = \frac{|q|B}{2\pi m}$, which is independent of the particle's speed or energy.
• Velocity with a Component Parallel to Field: If $\vec{v}$ has a component parallel to $\vec{B}$ ($v_\parallel$), this component remains unchanged. The component of velocity perpendicular to $\vec{B}$ ($v_\perp$) causes circular motion. The resulting path is a helix with radius $r = \frac{mv_\perp}{|q|B}$ and pitch $p = v_\parallel T = \frac{2\pi m v_\parallel}{|q|B}$.

Example 4.3: Calculates the radius of the circular path, frequency, and energy of an electron moving in a magnetic field.

Figure 4.5 illustrates circular motion, and Figure 4.6 shows helical motion.

Magnetic Field Due To A Current Element, Biot-Savart Law

The Biot-Savart Law describes the magnetic field ($d\vec{B}$) produced by an infinitesimal current element $I d\vec{l}$ at a point located by position vector $\vec{r}$ from the element:

$d\vec{B} = \frac{\mu_0}{4\pi} \frac{I (d\vec{l} \times \hat{r})}{r^2}$

where $\mu_0$ is the permeability of free space ($\mu_0 = 4\pi \times 10^{-7} \text{ T m/A}$), $I$ is the current, $d\vec{l}$ is the vector element of length along the current, $\vec{r}$ is the vector from the element to the point, and $\hat{r}$ is the unit vector in the direction of $\vec{r}$. The magnitude of the field is $dB = \frac{\mu_0}{4\pi} \frac{I |dl| \sin\theta}{r^2}$, where $\theta$ is the angle between $d\vec{l}$ and $\vec{r}$.

Key features compared to Coulomb's Law:

• Magnetic field sources are vector elements ($I d\vec{l}$), unlike scalar charges.
• Magnetic field is perpendicular to both the current element and the displacement vector.
• Magnetic field depends on the angle $\theta$.

The relationship between $\mu_0$, $\epsilon_0$, and the speed of light $c$ is $c = 1/\sqrt{\mu_0 \epsilon_0}$.

Example 4.4: Calculates the magnetic field produced by a small current element.

Magnetic Field On The Axis Of A Circular Current Loop

For a circular loop of radius $R$ carrying current $I$, the magnetic field at an axial point $P$ at a distance $x$ from the center is:

$\vec{B} = \frac{\mu_0 I R^2}{2(x^2 + R^2)^{3/2}} \hat{i}$

where $\hat{i}$ is the unit vector along the axis. At the center of the loop ($x=0$):

$\vec{B}_{center} = \frac{\mu_0 I}{2R} \hat{i}$

The magnetic moment of a current loop is defined as $\vec{m} = IA \hat{n}$, where $A$ is the area of the loop and $\hat{n}$ is the unit vector normal to the plane of the loop (direction given by the right-hand thumb rule).

Example 4.5: Calculates the magnetic field at the center of a semicircular arc.

Example 4.6: Calculates the magnetic field at the center of a tightly wound circular coil (multiple turns).

Figure 4.10 shows magnetic field lines for a current loop and the right-hand thumb rule for direction.

Ampere’s Circuital Law

Ampere's Circuital Law provides an alternative way to calculate magnetic fields, especially for situations with symmetry. It states that the line integral of the magnetic field $\vec{B}$ around any closed loop (Amperian loop) is equal to $\mu_0$ times the total current ($I_{enc}$) enclosed by the surface bounded by the loop:

$\oint_C \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$

This law is analogous to Gauss's Law in electrostatics. For symmetric situations, it simplifies to $BL = \mu_0 I_{enc}$, where $B$ is the magnitude of the magnetic field tangential to the loop of length $L$.

Example 4.7: Uses Ampere's Law to calculate the magnetic field inside and outside a long straight wire with a uniform current distribution.

Figure 4.12 illustrates Ampere's Circuital Law, and Figure 4.13 shows the geometry for calculating the field of a current-carrying wire using this law.

The Solenoid

A solenoid is a coil of wire wound into a helical shape. For a long solenoid (length much greater than its radius), the magnetic field inside is nearly uniform and directed along the axis. The magnetic field outside the solenoid is approximately zero.

The magnitude of the magnetic field inside a long solenoid is given by:

$B = \mu_0 n I$

where $n$ is the number of turns per unit length and $I$ is the current. Solenoids are crucial for creating uniform magnetic fields, and their strength can be significantly increased by inserting a soft iron core.

Figure 4.15 shows the magnetic field lines of a finite solenoid, and Figure 4.16 depicts the Amperian loop used to derive the field inside a long solenoid. Example 4.8 calculates the magnetic field inside a solenoid.

Force Between Two Parallel Currents, The Ampere

Two parallel conductors carrying currents exert magnetic forces on each other. If the currents are in the same direction, they attract each other. If the currents are in opposite directions, they repel each other. This force is the basis for defining the SI unit of current, the Ampere.

The force per unit length ($f$) between two long, parallel wires carrying currents $I_a$ and $I_b$, separated by a distance $d$, is:

$f = \frac{\mu_0}{2\pi} \frac{I_a I_b}{d}$

The Ampere is defined such that two parallel conductors, 1 meter apart in vacuum, carrying 1 Ampere each, experience a force of $2 \times 10^{-7}$ Newtons per meter of length.

Example 4.9: Calculates the force per unit length on a conductor carrying current in the Earth's magnetic field, highlighting the need for shielding in practical current measurements.

Torque On Current Loop, Magnetic Dipole

A rectangular current loop placed in a uniform magnetic field experiences a torque that tends to align the plane of the loop or its magnetic moment with the field. The torque is given by:

$\vec{\tau} = I (\vec{A} \times \vec{B})$

where $I$ is the current, $\vec{A}$ is the area vector of the loop (magnitude $A$, direction normal to the plane given by the right-hand thumb rule), and $\vec{B}$ is the magnetic field.

The magnetic moment ($\vec{m}$) of a current loop is defined as $\vec{m} = IA \hat{n}$, where $\hat{n}$ is the unit vector normal to the plane of the loop. For a coil with $N$ turns, $\vec{m} = NI\vec{A}$. The torque can then be written as:

$\vec{\tau} = \vec{m} \times \vec{B}$

This is analogous to the torque on an electric dipole in an electric field.

Circular Current Loop As A Magnetic Dipole

At large distances, the magnetic field produced by a circular current loop resembles the field of an electric dipole. The loop acts as a magnetic dipole with magnetic moment $\vec{m} = I\vec{A}$. The magnetic field on the axis of the loop at a large distance $x$ is approximately:

$\vec{B} \approx \frac{\mu_0}{2\pi} \frac{m}{x^3}$

This analogy highlights that current loops are the fundamental magnetic elements, unlike electric dipoles which are formed by two separate charges.

Example 4.10: Calculates the magnetic field at the center of a coil, its magnetic moment, the torque on it in an external field, and the angular speed acquired after rotation.

Example 4.11: Discusses the orientation of a current loop in a magnetic field for stable equilibrium and the shape a flexible wire assumes in a magnetic field.

The Moving Coil Galvanometer

A moving coil galvanometer (MCG) is an instrument used to detect and measure small currents. It operates based on the principle that a current-carrying coil in a uniform radial magnetic field experiences a torque ($\tau = NIAB \sin\theta$). This torque is balanced by a restoring torque from a spring ($\tau_{restore} = k\phi$), leading to a deflection proportional to the current: $\phi = (\frac{NAB}{k})I$.

• Current Sensitivity: Deflection per unit current ($\frac{NAB}{k}$).
• Voltage Sensitivity: Deflection per unit voltage ($\frac{NAB}{kR_G}$), where $R_G$ is the galvanometer resistance.

A galvanometer can be converted into an ammeter (to measure current) by connecting a low-resistance shunt resistor in parallel. It can be converted into a voltmeter (to measure voltage) by connecting a high-resistance resistor in series.

Figure 4.20 shows the schematic of an MCG, and Figures 4.21 and 4.22 illustrate its conversion to an ammeter and voltmeter, respectively. Example 4.12 calculates the current measured by a galvanometer, an ammeter (galvanometer with shunt), and an ideal ammeter.

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