# 29β1 Induced EMF #

- A changing magnetic field induces an emf
- That is, changing .$\vec B$, not .$\vec B$ itself, induces current
- Constant magnetic fields produce no current in a conductor
- This process is called
**electromagnetic induction**

- It doesn’t matter if the magnet or coil moves, only that there is
*relative*motion between the two

(a) A current is induced when a magnet is moved toward a coil, momentarily increasing the magnetic field through the coil. (b) The induced current is opposite when the magnet is moved away from the coil (.$\vec B$ decreases). In (c), no current is induced if the magnet does not move relative to the coil. It is the relative motion that counts here: the magnet can be held steady and the coil moved, which also induces an emf.

# 29β2 Faraday’s Law of Induction; Lenz’s Law #

- EMF is proportional to the rate of change of the magnetic flux .$\Phi_B$ passing through the circuit or loop with area .$A$
- Given a uniform magnetic field .$\vec B$ we write $$\Phi_B = B_\perp A = BA \cos\theta = \vec B \cdot \vec A$$

- For non-uniform fields, we need to integrate: $$\Phi_B = \int \vec B \cdot d \vec A$$

- The unit of magnetic flux is the tesla-meter, called the
**weber**: .$\text{1 Wb = 1 T$\cdot$ m$^2$}$ **Faraday’s law of induction**- The emf .$\mathscr{E}$ induced in a circuit is equal to the rate of change of magnetic flux through the circuit:
$$\mathscr{E} = -N\frac{d\Phi_B}{dt}$$
- .$N$ is the number of loops closely wrapped so the flux passes through each

- The emf .$\mathscr{E}$ induced in a circuit is equal to the rate of change of magnetic flux through the circuit:
$$\mathscr{E} = -N\frac{d\Phi_B}{dt}$$
**Lenz’s Law**- A current produced by an induced emf creates a magnetic field that
**opposes**the original change in magnetic flux- Faraday’s law is negative accordingly

- Realize that we know have to magnetic fields:
- The changing magnetic field or flux that induces the current, and
- The magnetic field produced by the induced current (all currents produce a magnetic field) which opposes the charge in the first

**Note:**The magnetic field created by induced current opposes change in external flux, not necessarily opposing the external field

- A current produced by an induced emf creates a magnetic field that
- It is important to note that an emf is induced whenever there is a change in flux through the coil – this can be done in three ways:
- Changing the magnetic field
- Changing the area .$A$ of the loop in the field
- Changing the loop’s orientation .$\theta$ wrt the field

## Problem Solving – Lenz’s Law #

Lenz’s law is used to determine the direction of the (conventional) electric current induced in a loop due to a change in magnetic flux inside the loop. (The loop may already be carrying its own ordinary current.) To produce an induced current you need (1) a closed conducting loop, and (2) an external magnetic flux through the loop that is changing in time.

- Determine whether the magnetic flux .$\Phi_B = \vec B \cdot \vec A$ inside the loop is decreasing, increasing, or unchanged.
- The magnetic field due to the induced current: (a) points in the same direction as the external field if the flux is
decreasing; (b) points in the opposite direction from the external field if the flux isincreasing; or (c) is zero if the flux isnot changing.- Once you know the direction of the induced magnetic field, use the right hand rule to find the direction of the induced current that would give this induced .$\vec B$
- Always keep in mind that there are two magnetic fields: (1) an external field whose flux must be changing if it is to induce an electric current, and (2) a magnetic field produced by the induced current.
- If a wire is already carrying an electric current, the total current while the magnetic field is changing will be the algebraic sum of the original current and the induced current.

# 29β3 EMF Induced in a Moving Conductor #

- If a conductor begins to move in a magnetic field, an emf is induced
- We can use Faraday’s law and kinematics to derive an equation:
$$\mathscr{E} = \frac{d\Phi_B}{dt} = \frac{B dA}{dt} = \frac{B (l\cdot v\ dt)}{dt} = Blv$$
- .$v$ is the speed of the conductor
- .$dA = l\ dx = lv\ dt$ is the change in area over time .$t$

- Be careful! This is a generalization where .$B, l, v$ are mutually perpendicular
- If they aren’t, we use the component of each that are mutually perpendicular

- We can use Faraday’s law and kinematics to derive an equation:
$$\mathscr{E} = \frac{d\Phi_B}{dt} = \frac{B dA}{dt} = \frac{B (l\cdot v\ dt)}{dt} = Blv$$
- An emf induced on a conductor moving ina magnetic field is sometimes called a motional emf
- We could also derive this using our force eq from ch27:
$$\vec F = q\vec v \times \vec B$$

- When the conductor moves with .$v$, as do its electrons
- Since .$\vec \perp \vec B$, each electron feels force .$F = qvB$
- The direction determined by the right hand rule (red)

- If the rod is not in contact with the conductor, electrons would collect at the upper end leaving the lower positive
- Therefore, there must be an induced emf!

- If the rod is in contact with the conductor, the electrons will flow into the conductor and there will be a clockwise current in the loop
- To calculate emf, we determine the work .$W$ needed to move a charge .$q$ from one end of the rod to the other against this potential difference: $$W = F \times d = (qvB) \times (l)$$ $$\mathscr{E} = W/q = qvBl/q = Blv$$

(a) A conducting rod is moved to the right on the smooth surface of a U-shaped conductor in a uniform magnetic field .$\vec B$ that points out of the page. The induced current is clockwise. (b) Force on an electron in the metal rod (moving to the right) is upward due to .$\vec B$ pointing out of the page. Hence electrons can collect at the top of the rod, leaving .$+$ charge at the bottom.

# 29β4 Electric Generators #

**Electric generators**produce electricity by transforming mechanical energy into electric energy- Also called
**dynamos** - Opposite of what a motor does

- Also called
- Consists of many wires wound around an
**armature**that can rotate in a magnetic field- This axel is turned by mechanical means (belt, steam, water falling, etc.) and an emf is induced in the rotating coil
- An ac current is thus the
*output*of a generator

An ac generator

- Each brush is fixed and presses against a
*continuous*slip ring that rotates with the armature - If the armature is rotating clockwise; then, .$\vec F = q \vec v \times \vec B$ applied to a charged particles in the wire
- Lenz’s law tells us that the (conventional) current in the wire .$b$ on the armature is outwards, towards us, thus, the current is outwards, through brush .$b$

- After one-half revolution, wire .$b$ will be where .$a$ is now and the current at .$b$ will be inwards. Thus, the current produced is alternating!
- If the loop is made to rotate in a uniform field .$\vec B$ with constant angular velocity .$\omega$, the emf produced is
$$\mathscr{E} = - \frac{d\Phi_B}{dt} = - \frac{d}{dt}\int \vec B \cdot d \vec A = -\frac{d}{dt}\big[BA\cos\theta\big]$$
- .$A$ is the area of the loop
- .$\theta$ is the angle between .$\vec B$ and .$\vec A$

- Since .$\omega = d\theta/dt \Longrightarrow \theta = \theta_0 + \omega t$, we choose .$\theta_0 = 0$ so
$$\mathscr{E} = - N BA \frac{d}{dt}(\cos(\omega t)) = N BA \omega \sin (\omega t)$$
- .$N$ is the number of loops in the rotating coil, assumed to be .$1$ unless stated otherwise
- Thus, the output wave is sinusoidal
- Amplitude .$\mathscr{E}_0 = NBA\omega$
- .$\mathscr{E}_\text{rms} = \frac{\mathscr{E}_0}{\sqrt{2}}$
- .$f = \omega / 2\pi$
- .$\text{60 Hz}$ for USA + Canada, .$\text{50 Hz}$ for EU

- dc generators
- Same for ac, except the slip rings are replaced by split-ring commutators (just like a dc motor)
- The curve output becomes more smooth by placing a capacitor in parallel with the output
- More commonly, is the use of many armature windings
- A capacitor tends to store charge and, if the time constant .$RC$ is long enough, helps to smooth out the voltage as shown in the figure to the right.

(a) A dc generator with one set of commutators, and (b) a dc generator with many sets of commutators and windings.

# 29β6 Transformers and Transmission of Power #

**Transformer:**Device used to increase or decrease an ac voltage- Made up of two coils know as the primary and secondary coil
- Primary is the input, secondary is the output
- These can be interwoven (with insulated wire); or can be linked by an iron core that’s laminated

- We assume energy losses (in resistance and hysteresis) can be ignored

- Made up of two coils know as the primary and secondary coil
- When an ac voltage is applied to the primary coil, the changing magnetic field it produces will induce an ac voltage of the same .$f$requency in the secondary coil
- However, secondary voltage, .$V_S$, changes according to the number of turns or loops in each coil:
$$V_S = N_S \frac{d\Phi_B}{dt}$$
- .$N_S$ is the number of turns in the secondary coil
- .$\Phi_B/dt$ is the rate at which the magnetic flux changes through each coil

- The input voltage, .$V_P$, is related to this rate too:
$$V_P = N_P \frac{d\Phi_B}{dt}$$
- .$N_P$ is the number of turns in the primary coil
- This follows because the changing flux produce a back emf, .$N_P\ d\Phi_B / dt$, in the primary that exactly balances the applied voltage .$V_P$
- This is because of Kirchhoff’s rules
- This is only the case if the resistance of the primary can be ignored (which we assume)

- Then, we can divide these two equations, assuming little or no flux loss, to find
$$\frac{V_S}{V_P} = \frac{N_S}{N_P}$$
- .$V_S$ and .$V_P$ can be the rms or peak values for both
- Step-up (.$N_S > N_P$) increase voltage; step-down (.$N_S < N_P$) decrease
- This is called the
**transformer equation**which tells us how the secondary (output) is related to the primary (input) - DC voltages don’t work in a transformer because there’d be no change in magnetic flux

- But muh conservation of energy!
- Power output is essentially the power input since transformers tend to be 99%+ efficient
- The little amount of energy lost is to heat

- Since .$P = IV$ and .$P_P \approx P_S$, we have $$I_P V_P = I_S V_S \Longrightarrow \frac{I_S}{I_P} = \frac{V_S}{V_P} = \frac{N_P}{N_S}$$

- Power output is essentially the power input since transformers tend to be 99%+ efficient

# 29β7 A Changing Magnetic Flux Produces an Electric Field #

**A changing magnetic flux produces an electric field**- This applies not only to wires and other conductors, but is a general result that applies to any region in space
- An electric field will be produced (induced) at any point in space where theres is a changing magnetic field
- These electric fields are not static, as are the electric fields due to electric charges at rest

## Faraday’s Law – General Form #

$$ \mathscr{E} = \oint \vec E \cdot d \vec l = - \frac{d\Phi_B}{dt}$$

- The first two terms come from the fact that the emf .$\mathscr{E}$ induced in a circuit is equal to the work done per unit charge by the electric field
- This is then combined with the relation of a changing magnetic flux to the the field it produces

## Forces Due to Changing .$\vec B$ are non-conservative #

- Electric field lines produced by a changing magnetic field are continuous; they form closed loops
- That is, while a conservative force (such as a electrostatic magnetic field) integrated over a line integral is zero .$\big(\oint \vec E_\text{electrostatic} \cdot d \vec l = 0\big)$, the electric field created by an magnetic field is not zero: .$\oint \vec E_\text{non-static} \cdot d \vec l = - \frac{d\Phi_B}{dt}$
- This implies that forces due to changing magnetic fields are non-conservative and we can’t define a potential energy (or even a potential function!) at a given point in space