Electromagnetic Induction Lecture Flashcards

Foundations of Electromagnetism and the Discovery of Induction

The study of electromagnetism establishes the fundamental behaviors of electric charges. To date, we have identified that electric charge creates electric fields, as described by Gauss’ Law, and moving electric charge creates magnetic fields, as described by Ampere’ Law. Michael Faraday ($1791 - 1867$) contributed significantly to this field. Notably, we understand through Gauss’ Law for Magnetic Fields that magnetic charge (monopoles) does not exist. In the $1830$’s, Michael Faraday proposed a deep relationship between the fields themselves, suggesting that an electric field could be created from magnetic fields. These created electric fields would subsequently produce a voltage in a wire and generate electric current. This specific voltage produced by magnetic fields is defined as induced emf ($ext{electromotive force}$), and the resulting currents are known as induced currents.

Experimental Evidence for Induced EMF

Faraday conducted experiments to test the creation of current from magnetic fields using an apparatus where one side generated a magnetic field and the other side measured current with an ammeter. He observed that while a steady current ran through the primary side, the ammeter on the secondary side recorded no current. Faraday concluded that a steady magnetic field is insufficient to produce a current; however, a changing magnetic field can successfully produce one. This phenomenon is known as electromagnetic induction. In his setup, the ammeter only measured current during the moments when the battery was being switched on or switched off, representing the intervals where the magnetic field was in flux. Visual data from these tests showed ammeter readings fluctuating between values such as $-2$, $0$, and $+2$ during these changes.

Faraday’s Law of Induction and Magnetic Flux

Faraday determined that the magnitude of the induced emf is proportional to the magnetic flux passing through the circuit. Magnetic flux is a measure of how much magnetic field passes through a given area. For a loop of wire, we define the magnetic flux ( ext{̦}) as follows:

̦ = \int \mathbf{B} \cdot d\mathbf{A}

The SI units for magnetic flux are Tesla-meters squared ($T \cdot m^2$), which is also called a weber ($\text{Wb}$). This value quantifies the total magnetic field penetrating an area. Faraday’s Law of Induction states that the amount of voltage produced is dependent on how quickly the flux changes through the loop. The formula is expressed as:

\mathcal{E} = -N \frac{d̦}{dt}

In this equation, $\mathcal{E}$ represents the induced emf in volts ($\text{V}$), $N$ is the number of loops, ̦ is the magnetic flux in webers ($\text{Wb}$), and $t$ is the time in seconds ($s$). The negative sign in the equation is an expression of Lenz’s Law, which states that the induced emf works to cancel out the change in magnetic flux in the loop. Specifically, a current produced by an induced emf moves in a direction so that the magnetic field created by that current opposes the original change in flux.

Example: Flux due to a Wire

Consider a wire holding a current $I_0$ placed near a square loop of side $l$. The loop is oriented such that its closest portion to the wire is at a distance $r$. Because the magnetic field strength decreases as we move away from the wire ($B = \frac{\mu_0 I_0}{2 \pi r}$), integration is required to find the total flux. Using the Right-Hand Rule (RHR), we determine the direction of the field passing through the loop. Using an area element $dA = l \, dr$, the integration is set up as:

̦ = \int_r^{r+l} \frac{\mu_0 I_0}{2 \pi r} \, l \, dr

̦ = \frac{\mu_0 I_0 l}{2 \pi} \int_r^{r+l} \frac{1}{r} \, dr

̦ = \frac{\mu_0 I_0 l}{2 \pi} (\ln(r+l) - \ln(r))

(Note: The transcript provides a different mathematical derivation step involving $-1/r^2$, but the goal is to find the total field through the geometric area of the square).

Visualization of Induction and Lenz's Law

A loop placed in a steady magnetic field creates no emf. If the magnetic field is increased, the flux through the loop increases (for example, creating more field lines into the page). As this flux changes, the loop induces an emf, creating an induced current in the counter-clockwise (CCW) direction. This induced current naturally sets up its own magnetic field (the induced magnetic field) directed out of the page. This induced field cancels out the change in flux, effectively attempting to maintain the original number of field lines. The Right-Hand Rule is the tool used to find the specific direction of the induced current based on whether the flux is increasing or decreasing.

In practical scenarios, if a circular loop is leaving a magnetic field, the flux is decreasing. To counteract this, the current flows counter-clockwise to increase the magnetic field through the loop. If a circular loop is shrinking within a magnetic field pointing into the page, the area is decreasing and flux is decreasing. In this case, the current will flow clockwise to increase the magnetic field through the loop.

Detailed Example: Faraday's Law Calculations

A square loop of wire with side $l = 5.0$ (interpreted as $0.50\,m$ in the calculation) is in a uniform magnetic field $B = 0.16\,T$.

a) When $B$ points directly out of the face of the loop ($\theta = 0^\circ$): ̦ = BA \cos(\theta) ̦ = (0.16\,T)(0.5\,m)^2 \cos(0^\circ) = 4.0 \times 10^{-4}\,\text{Wb}

b) When $B$ is at a $30^\circ$ angle to the area of the loop: ̦ = (0.16\,T)(0.5\,m)^2 \cos(30^\circ) = 3.5 \times 10^{-4}\,\text{Wb}

c) Induced emf as the loop rotates from $30^\circ$ to $0^\circ$ in $0.14\,s$: \mathcal{E} = - \frac{\Delta ̦}{\Delta t} = - \frac{\u0326_2 - ̦_1}{\Delta t} $\mathcal{E} = - \frac{(3.5 \times 10^{-4}\,\text{Wb}) - (4.0 \times 10^{-4}\,\text{Wb})}{0.14\,s} = 3.6 \times 10^{-4}\,\text{V}$

d) Current magnitude and direction if resistance R = 0.012\,̩: I = \frac{\mathcal{E}}{R} = \frac{3.6 \times 10^{-4}\,\text{V}}{0.012\,̩} = 0.030\,A Direction: Because the loop is losing flux as it spins, the induced current spins in a direction to add the flux back.

Example: Pulling a Square Coil

A $100$-loop square of wire with side $l = 5.00\,cm$ ($0.0500\,m$) and total resistance R = 100\,̩ is positioned perpendicular to a uniform magnetic field $B = 0.600\,T$. It is pulled from the field at a constant speed, reaching a field-free region in $0.100\,s$.

a) Rate of change in flux for one loop: \frac{\Delta ̦}{\Delta t} = \frac{0 - (0.600\,T)(0.0500\,m)^2}{0.100\,s} = -1.50 \times 10^{-2}\,\text{Wb/s}

b) Total emf induced and current in the $100$-loop coil: \mathcal{E} = -N \frac{\Delta ̦}{\Delta t} = -(100)(-1.50 \times 10^{-2}\,\text{Wb/s}) = 1.50\,V I = \frac{\mathcal{E}}{R} = \frac{1.50\,V}{100\,̩} = 0.0150\,A = 15.0\,mA

c) Energy dissipated in the coil: $P = I\mathcal{E} = (0.0150\,A)(1.50\,V) = 0.0225\,W$ $E = P \Delta t = (0.0225\,W)(0.100\,s) = 2.25 \times 10^{-3}\,J$

d) Force required to pull the loop out of the field: Pulling the loop induces a magnetic field and a resulting magnetic force that pulls it back. This force $F$ can be found via the energy/power relationship $P = Fv$ or $F = E/d$: $F = \frac{E}{l} = \frac{2.25 \times 10^{-3}\,J}{0.0500\,m} = 0.0450\,N$

Motional EMF

Emf can also be induced in a moving conductor. This is demonstrated by a setup where a conducting rod is dragged to the right at a speed $v$ while in contact with a U-shaped wire. As the rod moves, the area within the loop increases, causing a flux change: \mathcal{E} = \frac{\Delta ̦}{\Delta t} = \frac{B \, \Delta A}{\Delta t} = \frac{B \, l \, \Delta x}{\Delta t} = Blv This is referred to as motional emf. For a rod with non-uniform motion defined by $x(t) = v_0 t^3$, the velocity is found by differentiating: $v(t) = \frac{dx}{dt} = 3v_0 t^2$. The induced emf as a function of time is then: $\mathcal{E} = Blv = B l (3v_0 t^2)$

Electric Generators: AC and DC

Induction is the primary method for creating alternating current ($AC$). This is achieved via a dynamo, which converts mechanical energy into electrical energy. An AC generator rotates a wheel (using steam from nuclear power or falling water from dams) within a magnetic field. The wheel contains a coil of wires, inducing an emf and resulting current. Slip rings allow for continuous contact with the rotating wire, producing an alternating current because the coil moves up and down on each side of the rotation. As the area increases relative to the field, current flows CW; as it decreases, it flows CCW.

If the wheel spins at an angular velocity $\omega$, the angle changes as $\theta = \omega t$. The voltage is: $\mathcal{E} = -N \frac{d}{dt} (BA \cos(\omega t)) = NBA\omega \sin(\omega t)$ Using multiple loops increases the voltage. The root mean square ($rms$) voltage is defined as: $V_{rms} = \frac{V_0}{\sqrt{2}}$

Direct current ($DC$) is produced by replacing slip rings with a split-ring commutator. The split-ring prevents continuous contact, ensuring the terminal only takes current in one direction. When the current would naturally flip direction during rotation, the split-ring connects to the opposite terminal to keep the current unidirectional. Multiple loops and capacitors with large time constants are utilized to smooth the resulting current.

Eddy Currents and Magnetic Damping

Induction occurs in bulk objects as well as wires. When a bulk piece of metal moves through an external magnetic field, Faraday’s Law applies to the area exposed to the field. Electrons in the metal move in circular paths to counter the magnetic field. These are called eddy currents. They are present in any conductor moving across a magnetic field where flux is changing. The motion of these currents creates a force that opposes the movement of the metal, a phenomenon known as magnetic damping.

Transformers

Transformers are induction devices used to increase or decrease AC voltage. They consist of a primary coil and a secondary coil. Flux produced by the primary coil passes through the secondary coil. The relationship between the voltages and the number of loops is defined by the transformer equation: $\frac{V_S}{V_P} = \frac{N_S}{N_P}$

Example: A cell phone wall charger reduces $120\,V$ AC from an outlet to $5.0\,V$ AC to charge a $3.7\,V$ battery. The secondary coil has $N_S = 30$ turns and supplies $I_S = 700\,mA$.

• Primary turns: $N_P = N_S (\frac{V_P}{V_S}) = 30 (\frac{120}{5}) = 720\,\text{turns}$

• Primary current: $I_P = I_S (\frac{N_S}{N_P}) = (0.700\,A) (\frac{30}{720}) = 29.0\,mA$

• Power transformed: $P = I_S V_S = (0.700\,A)(5.0\,V) = 3.50\,W$

Faraday’s Law: Relation to Electric Fields

Faraday’s law can be generalized to show the explicit relationship between electric and magnetic fields. Potential difference is defined as a loop integral of the electric field ($\oint \mathbf{E} ∙ d\mathbf{l}$). A changing magnetic field produces an electric field: \oint \mathbf{E} ∙ d\mathbf{l} = - \frac{d̦}{dt} The direction of the electric field follows the direction of the induced current (determined by RHR).

If a uniform magnetic field of radius $R$ changes at a constant rate $dB/dt$, the electric field at distance $r_0$ (r_0 < R) is found by integrating over a loop of radius $r_0$: $E(2\pi r_0) = \frac{d}{dt} (B \pi r_0^2) = \pi r_0^2 \frac{dB}{dt}$ $E = \frac{r_0}{2} \frac{dB}{dt}$

At a point $r_1$ (r_1 > R) outside the magnetic field area, the magnetic field stops at $R$, so the flux integration is limited to the radius of the material region: $E(2\pi r_1) = \pi R^2 \frac{dB}{dt}$ $E = \frac{R^2}{2r_1} \frac{dB}{dt}$