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Unit 12: Magnetic Fields and Electromagnetism

This unit covers the behavior of magnetic fields and their relationship with electric currents, charges in motion, and changing electric fields. Students will study how magnetic forces act on moving charges and current-carrying wires, apply the Biot–Savart Law and Ampère’s Law to calculate magnetic fields, and analyze the interplay between electricity and magnetism in Faraday’s Law of Induction. Understanding magnetic flux, inductance, and electromagnetic energy storage is essential for solving complex problems involving generators, transformers, and AC/DC circuit components. This unit builds on earlier topics by integrating vector calculus and symmetry principles to describe the unified nature of electromagnetic phenomena.

Velocity Selector (Crossed \( \vec{E} \) and \( \vec{B} \) Fields)

In crossed-field devices, a uniform electric field \( \vec{E} \) and magnetic field \( \vec{B} \) are arranged perpendicular to each other and to the incoming beam. A charged particle experiences electric force \( q\vec{E} \) and magnetic force \( q\,\vec{v}\times\vec{B} \) in opposite directions; only particles with speed \( v = \frac{E}{B} \) feel balanced forces and pass straight. Others are deflected up or down, so the apparatus “selects” a narrow speed band.
This principle appears in mass spectrometers and e/m determinations, where the selector ensures a uniform entry speed before subsequent bending in a known magnetic field. An immediate check on set-up is units: \( E/B \) has units \( (\text{N/C})/(\text{T}) = \text{m/s} \), confirming it represents a speed. When solving, track the sign of \( q \) to determine the actual deflection direction.

Forces on Collections of Charge (Beams and Plasmas)

For a narrow, monoenergetic beam, you can analyze a representative charge to determine the entire beam’s path because all particles share the same \( q/m \) and speed. In practice, small spreads in \( v \) cause slight radius differences \( r=\frac{mv}{|q|B} \), widening the beam downstream. Collisions or non-uniform fields can further broaden distributions, which is why focusing elements (magnetic lenses) are often used.
In plasmas where both signs of charge are present, magnetic forces bend ions and electrons in opposite senses due to the sign of \( q \). Their very different masses also produce vastly different gyroradii and cyclotron frequencies, separating species. Understanding these scalings is crucial in fusion confinement and in space physics, where natural \( \vec{B} \) fields guide particle motion.

Magnetic Forces on Current-Carrying Wires

Force on a Straight Current in a Uniform Field

The magnetic force on a segment of wire carrying current \( I \) in a magnetic field \( \vec{B} \) is given by \( \vec{F} = I\,\vec{L} \times \vec{B} \), where \( \vec{L} \) points along the current with magnitude equal to the straight-line length of the segment. The right-hand rule sets the direction: fingers along current, curl toward \( \vec{B} \), thumb gives \( \vec{F} \). Because the cross product depends on \( \sin\theta \), there is no force when the wire is parallel to \( \vec{B} \) and maximum force when it is perpendicular.
This force acts on the charges in the conductor and is transmitted to the lattice, so the whole wire experiences a mechanical push. In practice, straight conductors can deflect, stretch supports, or vibrate at AC frequencies due to time-varying directions of \( \vec{F} \). Lab setups often exploit this to measure \( B \) by weighing the wire and looking at the change in apparent weight when current flows.
Nonuniform fields produce different forces on different portions of the wire, requiring an integral \( \vec{F} = I \int d\vec{\ell} \times \vec{B} \). This matters near magnets or inside devices where \( \vec{B} \) varies with position, such as near the pole faces of motors. When solving, break the wire into simple pieces where \( \vec{B} \) is known and sum vector contributions carefully.

Force Between Parallel Currents

Two long, straight, parallel wires exert magnetic forces on each other because each wire’s current creates a magnetic field at the location of the other. For wire 1 creating \( B_1 = \dfrac{\mu_0 I_1}{2\pi r} \) at wire 2 a distance \( r \) away, the force per unit length on wire 2 is \( \dfrac{F_{21}}{L} = I_2 B_1 = \dfrac{\mu_0 I_1 I_2}{2\pi r} \). The direction follows the right-hand rule: currents in the same direction **attract**, while currents in opposite directions **repel**.
This interaction defines the SI ampere historically and underpins practical issues like busbar spacing in power systems. The inverse dependence on \( r \) means forces can be large at small separations, so mechanical supports must withstand magnetic loads during high-current operation. On exams, draw field circles around one wire to see the field direction at the other, then apply \( \vec{F} = I\vec{L}\times\vec{B} \) to get attraction or repulsion.
Superposition applies when more than two wires are present: compute the net \( \vec{B} \) at each wire from all the others, then find that wire’s force. Symmetric arrangements (e.g., three-phase conductors) often simplify due to geometric cancellations. Always keep current directions and relative positions straight—one reversed arrow can flip an attraction to a repulsion in your reasoning.

Distributed Currents and Non-Straight Conductors

For curved conductors like arcs, each infinitesimal element \( d\vec{\ell} \) experiences \( d\vec{F} = I\, d\vec{\ell} \times \vec{B} \), and the total force is the vector integral over the path. In uniform fields, symmetry can make some components cancel (e.g., opposite sides of a rectangular loop), leaving a net torque but small net force. This is the stepping stone to understanding why loops rotate in motors while their centers may not translate much.
When the field is produced by coils or magnets with structure, the direction of \( \vec{B} \) can change across the conductor’s span. Approximating \( \vec{B} \) as piecewise uniform or using known field expressions (e.g., inside long solenoids) keeps the math manageable. Careful sketches with \( \vec{B} \), current direction, and element orientation prevent sign mistakes in the line integral.
Edge effects and lead wires can contribute forces that alter the net result in real devices, especially when currents are high. Engineers minimize unintended forces by routing returns close to feeds, reducing loop area and the corresponding magnetic interactions. This same idea appears in EM induction topics where loop area controls flux linkage and induced effects.

Biot–Savart Law

Definition and Formula

The Biot–Savart Law calculates the magnetic field \( \vec{B} \) generated at a point in space by a small segment of current-carrying conductor. The differential form is \( d\vec{B} = \frac{\mu_0}{4\pi} \frac{I\, d\vec{\ell} \times \hat{r}}{r^2} \), where \( d\vec{\ell} \) points along the current, \( r \) is the distance from the element to the field point, and \( \hat{r} \) is the unit vector from the element to that point. The cross product indicates that only the component of \( d\vec{\ell} \) perpendicular to \( \hat{r} \) contributes to \( \vec{B} \).
This law is most useful for computing \( \vec{B} \) in situations lacking high symmetry, where Ampère’s Law would be difficult to apply. For example, finite straight wires, current arcs, or complex geometries can be handled by integrating the Biot–Savart expression along the current path. The resulting field is the vector sum of contributions from each differential element.
The Biot–Savart Law links directly to Coulomb’s Law in form, but with moving charges and a perpendicular relationship between current direction and the field. It emphasizes that \( \vec{B} \) decreases with the square of the distance and wraps in circles around the current element, as shown by the right-hand rule. This microscopic view builds a foundation for understanding macroscopic magnetic fields from current distributions.

Applications and Examples

A common application is finding \( \vec{B} \) at the center of a circular loop: symmetry makes each \( d\vec{\ell} \) contribute equally in magnitude and direction, leading to \( B = \frac{\mu_0 I}{2R} \). This is a stepping stone to analyzing solenoids and toroids as sums of many loops. Such calculations also guide coil design for uniform fields in laboratory equipment.
For finite straight conductors, the Biot–Savart Law gives \( B = \frac{\mu_0 I}{4\pi R} (\sin\theta_1 + \sin\theta_2) \), where \( \theta_1 \) and \( \theta_2 \) define the angular extent from the observation point. This is particularly useful for determining fields near edges of wires or busbars, where the infinite-wire formula would overestimate \( B \).
In exam problems, symmetry arguments can drastically simplify the integral. For example, at the center of a square loop, each side produces a field of equal magnitude, and vector addition yields the net \( \vec{B} \). Remember that changing the observation point can break symmetry, forcing a full Biot–Savart integration instead of shortcuts.

Connection to Ampère’s Law

The Biot–Savart Law is more general than Ampère’s Law because it does not require symmetry, but it is often harder to compute with due to its integral over the current path. Ampère’s Law emerges naturally from Biot–Savart when applied to highly symmetric configurations such as infinite wires or solenoids. Recognizing when symmetry allows the switch from Biot–Savart to Ampère saves time on both exams and real-world engineering problems.
Both laws reflect the principle of superposition, meaning the net field from multiple current sources is the sum of the fields from each source. This mirrors electrostatics with multiple charges and Coulomb’s Law. Developing fluency in toggling between Biot–Savart and Ampère’s Law depending on geometry is a key problem-solving skill in electromagnetism.
Because Biot–Savart explicitly integrates over current elements, it also connects directly to the concept of magnetic vector potential \( \vec{A} \), which is central in more advanced electromagnetism and quantum contexts. This reinforces the idea that magnetism arises from moving charges and is inseparable from electric phenomena under relativity.

Ampère’s Law

Definition and Formula

Ampère’s Law relates the line integral of the magnetic field \( \vec{B} \) around a closed loop to the total current \( I_{\text{enc}} \) enclosed by that loop. Mathematically, it is written as \( \oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{\text{enc}} \). This law applies exactly in magnetostatics and is a direct consequence of the Biot–Savart Law when applied to symmetric current configurations.
The law is especially powerful when the geometry of the current distribution has high symmetry—cylindrical, planar, or toroidal. In such cases, the magnetic field’s magnitude is constant along the chosen Amperian loop, which allows \( B \) to be factored out of the integral, greatly simplifying the calculation. This turns the line integral into \( B(2\pi r) = \mu_0 I_{\text{enc}} \) for an infinite straight wire, for example.
Ampère’s Law is not limited to straight wires—it also applies to solenoids and toroids. For a solenoid, \( B = \mu_0 n I \), where \( n \) is the number of turns per unit length. For a toroid, \( B = \frac{\mu_0 N I}{2\pi r} \), where \( N \) is the total number of turns and \( r \) is the radial distance from the center. These results follow directly from applying the law to an Amperian path inside the device.

Applications and Examples

For an infinitely long straight wire carrying current \( I \), symmetry dictates that \( \vec{B} \) circles the wire uniformly. Applying Ampère’s Law to a circular path of radius \( r \) gives \( B = \frac{\mu_0 I}{2\pi r} \). This is identical to the infinite-wire Biot–Savart result, showing the two laws are consistent in symmetric cases.
Inside a long solenoid with \( n \) turns per unit length and current \( I \), Ampère’s Law yields \( B = \mu_0 n I \). Outside the solenoid, symmetry arguments and the negligible external field mean \( B \) is effectively zero. This explains why solenoids are used to create nearly uniform fields for experiments.
For a toroidal coil, the path length in Ampère’s Law is \( 2\pi r \), leading to \( B = \frac{\mu_0 N I}{2\pi r} \) inside the coil and zero outside. This design confines the magnetic field, which is valuable in minimizing interference in surrounding electronics.

Connections and Limitations

While Ampère’s Law is exact, its direct application is practical only when symmetry allows simplification. In asymmetric situations, the integral becomes as challenging as Biot–Savart, and numerical methods or approximations are required. Recognizing symmetry quickly is key to solving problems efficiently.
Ampère’s Law connects deeply to Maxwell’s equations, forming one of the four core laws of electromagnetism. In the time-varying case, it is modified to include Maxwell’s displacement current term, which allows it to describe changing electric fields as sources of magnetic fields—crucial for understanding electromagnetic waves.
Conceptually, Ampère’s Law mirrors Gauss’s Law in electrostatics: Gauss’s Law relates electric flux to enclosed charge, while Ampère’s Law relates the circulation of \( \vec{B} \) to enclosed current. Both are integral forms of more general field equations and reflect the symmetry-based power of vector calculus in physics.

Magnetic Flux

Definition and Formula

Magnetic flux measures the total magnetic field \( \vec{B} \) passing through a given surface area \( A \). It is mathematically defined as \( \Phi_B = \int \vec{B} \cdot d\vec{A} \), where \( d\vec{A} \) is an infinitesimal vector area element pointing perpendicular to the surface. The dot product accounts for the fact that only the component of \( \vec{B} \) perpendicular to the surface contributes to the flux.
The SI unit of magnetic flux is the weber (Wb), where \( 1 \ \mathrm{Wb} = 1 \ \mathrm{T \cdot m^2} \). A single weber corresponds to one tesla of magnetic field uniformly passing through an area of one square meter, oriented perfectly perpendicular to the field.
Magnetic flux is a scalar quantity, but its sign can be positive or negative depending on the orientation of the area vector relative to the magnetic field. A positive flux means \( \vec{B} \) is in the same direction as \( d\vec{A} \), while a negative flux means they point in opposite directions.

Physical Meaning

Physically, magnetic flux represents how much magnetic field “passes through” a surface. A larger magnetic field strength, a larger area, or a more perpendicular alignment between \( \vec{B} \) and the surface increases the flux. If the surface is tilted, the flux decreases according to the cosine of the tilt angle.
Flux is a central concept in Faraday’s Law of Induction, where a changing magnetic flux induces an electromotive force (EMF) in a closed conducting loop. This is the basis for electric generators and many electromagnetic devices.
For closed surfaces, Gauss’s Law for Magnetism states that the total magnetic flux is always zero, \( \oint \vec{B} \cdot d\vec{A} = 0 \), reflecting the fact that magnetic monopoles have never been observed. Magnetic field lines form continuous loops with no start or end point.

Examples and Applications

In a uniform magnetic field of strength \( B \) perpendicular to a flat surface of area \( A \), the flux is simply \( \Phi_B = B A \). If the same surface is tilted by angle \( \theta \) relative to \( \vec{B} \), the flux becomes \( \Phi_B = B A \cos\theta \).
In electric generators, the rotation of a coil in a constant magnetic field causes the angle \( \theta \) to change with time, producing a sinusoidal variation in magnetic flux. According to Faraday’s Law, this results in an alternating current (AC) output.
Magnetic resonance imaging (MRI) machines use large magnetic flux values to align atomic nuclei in the body. Changing the flux with radiofrequency pulses allows for precise imaging of tissues based on their magnetic response.

Connections to Other Topics

Magnetic flux is conceptually similar to electric flux from Gauss’s Law in electrostatics. Both describe how much of a field passes through a surface and are used in symmetry-based field calculations.
Flux links magnetostatics to electromagnetic induction. Understanding flux now prepares you for Faraday’s and Lenz’s Laws, where the time rate of change of flux determines the induced EMF in a conductor.
Magnetic flux is also closely tied to the concept of field lines: a higher density of field lines passing through a surface corresponds to a greater magnitude of flux, making it an intuitive bridge between diagrams and equations.

Faraday’s Law of Induction

Definition and Formula

Faraday’s Law states that a changing magnetic flux through a closed conducting loop induces an electromotive force (EMF) in the loop. Mathematically, it is expressed as \( \mathcal{E} = -\frac{d\Phi_B}{dt} \), where \( \Phi_B \) is the magnetic flux and the negative sign represents Lenz’s Law, indicating that the induced EMF opposes the change in flux.
This law is a fundamental principle of electromagnetism, explaining how electric currents can be generated from magnetic fields. It underlies the operation of generators, transformers, and inductive sensors.
In practical terms, the EMF can drive a current in the loop if a conductive path is present, converting mechanical work (such as moving a magnet) into electrical energy.

Key Concepts

A change in magnetic flux can be produced by varying the magnetic field strength, the area of the loop, or the orientation of the loop relative to the field. Any of these factors can induce an EMF according to Faraday’s Law.
The negative sign in the equation is essential for energy conservation — the induced current always flows in a direction that creates a magnetic field opposing the change in the original flux. This is Lenz’s Law in action.
Faraday’s Law applies equally to loops of wire that are stationary in a changing magnetic field and to loops moving through a spatially varying field. Both situations involve a time-dependent change in flux.

Motional EMF

Definition and Formula

Motional EMF occurs when a conductor moves through a magnetic field, causing free charges in the conductor to experience a magnetic force. This force separates charges along the conductor, creating a potential difference. The formula for motional EMF is \( \mathcal{E} = B \ell v \sin\theta \), where \( B \) is the magnetic field strength, \( \ell \) is the length of the conductor in the field, \( v \) is its velocity, and \( \theta \) is the angle between \( \vec{v} \) and \( \vec{B} \).
This induced EMF arises from the Lorentz force \( \vec{F} = q(\vec{v} \times \vec{B}) \) acting on charges within the moving conductor. Positive and negative charges are pushed in opposite directions, leading to an electric field that opposes further charge separation.
Motional EMF is a specific case of Faraday’s Law where the change in magnetic flux comes from physically moving a conductor through a magnetic field rather than changing the field itself.

Key Concepts

If the conductor moves perpendicular to the magnetic field (\( \theta = 90^\circ \)), the EMF is maximized. If it moves parallel to the field (\( \theta = 0^\circ \)), no EMF is induced because no magnetic force acts on the charges.
In circuits containing a moving conductor, the induced EMF can drive a current if a complete path exists. This current can, in turn, create its own magnetic field, which opposes the motion according to Lenz’s Law.
Real-world applications of motional EMF include electric generators, railguns, and electromagnetic braking systems, where mechanical motion is converted into electrical energy or vice versa.

Lenz’s Law

Definition

Lenz’s Law states that the direction of the induced current in a closed loop is always such that the magnetic field it produces opposes the change in magnetic flux that induced the current. This is mathematically represented by the negative sign in Faraday’s Law: \( \mathcal{E} = -\frac{d\Phi_B}{dt} \).
This opposition is not arbitrary — it is required by the law of conservation of energy. If the induced current supported the change instead of opposing it, it would create a self-reinforcing effect that violates energy conservation.
Lenz’s Law applies to both stationary loops in a changing magnetic field and moving conductors in a constant field, making it fundamental in understanding induced EMF in all scenarios.

Key Concepts

The direction of the induced current can be determined using the right-hand rule: point your thumb in the direction of the induced magnetic field that opposes the change, and your fingers curl in the direction of the induced current.
If the magnetic flux through a loop is increasing, the induced current creates a magnetic field in the opposite direction to counteract the increase. If the flux is decreasing, the induced field is in the same direction as the original to resist the decrease.
Understanding Lenz’s Law is crucial for predicting the behavior of systems like electric generators, inductors, and electromagnetic brakes, all of which rely on induced currents that oppose the cause of induction.

Inductance and Self-Induction

Definition of Inductance

Inductance is the property of a circuit or component that causes it to oppose changes in current by inducing an EMF. It is denoted by \( L \) and measured in henrys (H). The induced EMF is given by \( \mathcal{E} = -L \frac{dI}{dt} \), where \( \frac{dI}{dt} \) is the rate of change of current.
A coil or solenoid has inductance because a changing current creates a changing magnetic field, which in turn changes the magnetic flux through the coil itself, leading to an induced EMF that opposes the change in current.
Inductance depends on the coil's geometry (number of turns, cross-sectional area, length) and the magnetic permeability of the core material.

Self-Induction

Self-induction occurs when a changing current in a circuit induces an EMF in the same circuit. This happens in inductors, where the changing magnetic field produced by the coil's own current induces a voltage opposing the change.
When current increases, the induced EMF opposes the increase; when current decreases, the induced EMF tries to maintain the current. This is a direct consequence of Lenz’s Law.
Self-induction is a key factor in the transient behavior of circuits, especially in RL and RLC circuits, where the rate at which current changes is limited by the inductor’s opposition.

Energy Stored in an Inductor

Inductors store energy in their magnetic fields. The energy stored is given by \( U = \frac{1}{2} L I^2 \), where \( I \) is the current through the inductor.
This stored magnetic energy can be released back into the circuit when the current changes, playing a role in oscillatory and transient phenomena.
Understanding energy storage in inductors is essential for analyzing power transfer and losses in AC and DC circuits involving inductive components.

Mutual Induction and Transformers

Definition of Mutual Induction

Mutual induction occurs when a changing current in one coil induces an EMF in another nearby coil due to the changing magnetic flux linking both coils. The amount of induced EMF depends on the mutual inductance \( M \), which is a measure of the coupling between the coils.
The induced EMF in the secondary coil is given by \( \mathcal{E}_2 = -M \frac{dI_1}{dt} \), where \( I_1 \) is the current in the primary coil. Similarly, the primary coil can experience an induced EMF due to current changes in the secondary coil.
Mutual inductance depends on the coils’ geometry, number of turns, distance, and the medium between them. Tight coupling and a ferromagnetic core increase \( M \).

Transformers

A transformer is a device that uses mutual induction to change the voltage of an alternating current. It consists of two coils — the primary and secondary — wound around a common core to maximize magnetic coupling.
The voltage relationship is given by \( \frac{V_s}{V_p} = \frac{N_s}{N_p} \), where \( N_s \) and \( N_p \) are the number of turns in the secondary and primary coils, respectively. This allows step-up or step-down voltage conversion.
While ideal transformers conserve power (\( P_p = P_s \)), real transformers experience energy losses due to resistance, eddy currents, and hysteresis in the core material.

Applications

Mutual induction is used in wireless charging, where energy is transferred between coils without physical contact, and in many communication devices that rely on inductive coupling.
Transformers are critical in the electrical grid, enabling efficient long-distance transmission of electricity by stepping up voltage to reduce current (and thus resistive losses) and then stepping it down for safe residential or industrial use.
Understanding mutual induction principles is essential for analyzing AC circuit behavior, impedance matching, and signal isolation in electronic systems.

RL Circuits

An RL circuit consists of a resistor (R) and an inductor (L) connected in series or parallel, often with a voltage source. Inductors oppose changes in current, causing the current to increase or decrease gradually instead of instantaneously.
When the circuit is connected to a voltage source, the current grows according to \( I(t) = I_{\text{max}} \left(1 - e^{-t/\tau} \right) \), where \( \tau = \frac{L}{R} \) is the time constant. This describes the charging phase of the inductor.
When the source is removed, the current decays as \( I(t) = I_0 e^{-t/\tau} \). The time constant determines how quickly the circuit responds to changes, which is important in timing applications and signal processing.

Energy Stored in Magnetic Fields

Inductors store energy in the magnetic fields created by the current flowing through them. The stored energy is given by \( U = \frac{1}{2} L I^2 \), where \( L \) is the inductance and \( I \) is the current.
This energy is not lost while the current is steady; it is released back into the circuit when the current decreases, allowing inductors to smooth out fluctuations in current flow.
In electromagnet design, maximizing the stored magnetic energy is important for applications such as transformers, motors, and electromagnetic launchers.

Magnetic Field of a Long Straight Current-Carrying Wire

A long, straight wire carrying current \( I \) generates a magnetic field that forms concentric circles around the wire, following the right-hand rule. The direction of the field depends on the direction of current flow.
Using Ampère’s Law, the field magnitude is \( B = \frac{\mu_0 I}{2\pi r} \), where \( r \) is the distance from the wire and \( \mu_0 \) is the permeability of free space. This relationship shows that the magnetic field decreases with distance.
This principle underlies the operation of many devices, including power transmission lines, electromagnets, and magnetic sensors that detect current flow.

Practice Problems

Problem 1: RL Circuit Charging

A 12.0 V battery is connected in series with a 4.0 Ω resistor and a 2.0 H inductor. Determine the current in the circuit after 1.5 s. Use \( I(t) = I_{\text{max}} \left(1 - e^{-t/\tau} \right) \) where \( \tau = \frac{L}{R} \).

Solution:

Step 1: Calculate the time constant: \( \tau = \frac{L}{R} = \frac{2.0}{4.0} = 0.50 \ \text{s} \).
Step 2: Determine \( I_{\text{max}} = \frac{V}{R} = \frac{12.0}{4.0} = 3.0 \ \text{A} \).
Step 3: Substitute into the charging equation: \( I(1.5) = 3.0 \left(1 - e^{-1.5 / 0.50} \right) \).
Step 4: Simplify: \( e^{-3} \approx 0.0498 \), so \( I(1.5) \approx 3.0 \times (1 - 0.0498) \approx 2.85 \ \text{A} \).
Final Answer: \( I \approx 2.85 \ \text{A} \) after 1.5 s.

Problem 2: Magnetic Field from a Long Straight Wire

A long straight wire carries a current of 8.0 A. Calculate the magnetic field strength at a distance of 0.020 m from the wire. Use \( B = \frac{\mu_0 I}{2\pi r} \) and \( \mu_0 = 4\pi \times 10^{-7} \ \text{T·m/A} \).

Solution:

Step 1: Write the formula: \( B = \frac{\mu_0 I}{2\pi r} \).
Step 2: Substitute values: \( B = \frac{(4\pi \times 10^{-7})(8.0)}{2\pi (0.020)} \).
Step 3: Cancel \( \pi \) and simplify: \( B = \frac{(4 \times 10^{-7})(8.0)}{0.040} \).
Step 4: \( B = \frac{3.2 \times 10^{-6}}{0.040} = 8.0 \times 10^{-5} \ \text{T} \).
Final Answer: \( B = 8.0 \times 10^{-5} \ \text{T} \) (80 µT).

Common Misconceptions

RL Circuits Reach Steady State Instantly: Many students assume that when a switch is closed in an RL circuit, the current immediately reaches its maximum value. In reality, the inductor resists changes in current due to self-induction, causing the current to increase gradually according to \( I(t) = I_{\text{max}}(1 - e^{-t/\tau}) \). This time delay is a direct result of energy being stored in the magnetic field of the inductor.
Inductors Always “Block” Current: Some think inductors completely stop current when the circuit is turned on. This is false—an inductor only resists changes in current at first. Over time, in steady-state DC, an ideal inductor acts like a wire with no resistance, meaning it allows full current flow.
Energy in a Magnetic Field Comes from Nowhere: Students sometimes believe magnetic fields inherently store energy without a source. In truth, the energy stored in the magnetic field of an inductor (\( U = \frac{1}{2} L I^2 \)) comes from the work done by the power source to build up the current against the induced EMF.
Field Around a Long Straight Wire is Uniform: A common error is thinking the magnetic field has the same strength everywhere around the wire. In reality, the field decreases with distance according to \( B = \frac{\mu_0 I}{2\pi r} \), so doubling the distance from the wire halves the field strength.
Direction of Induced Fields is Arbitrary: Some students think they can choose any direction for the induced magnetic field in Lenz’s Law problems. The direction is not arbitrary—it is determined by the requirement to oppose the change in flux, preserving the conservation of energy.