EMF in a straight conductor: $\varepsilon = BLv$

Electromagnetic Induction — AQA A-Level Physics

When a straight conductor of length L moves at velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹. v perpendicular to a uniform magnetic field B, an EMFElectromotive force. The energy transferred per unit charge by a source in driving charge around a complete circuit. Measured in volts (V). is induced across its ends.
This is a direct application of Faraday's lawThe magnitude of the induced EMFElectromotive force. The energy transferred per unit charge by a source in driving charge around a complete circuit. Measured in volts (V). is proportional to the rate of change of magnetic flux linkageThe product of magnetic flux and the number of turns of a coil. Measured in weberThe SI unit of magnetic flux. One weber is the flux through an area of 1 m² when the magnetic flux density is 1 T perpendicular to the area.-turns (Wb turns)... As the wire moves, it sweeps out area. That area sits inside the magnetic field, so the flux through the circuit changes.

\varepsilon = BLv

$ε$: induced EMFElectromotive force. The energy transferred per unit charge by a source in driving charge around a complete circuit. Measured in volts (V). (V)
$B$: magnetic flux densityMass per unit volume of a material. Measured in kg m⁻³.The strength of a magnetic field. The force per unit length per unit currentThe rate of flow of chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C).. Measured in amperes (A). on a currentThe rate of flow of chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C).. Measured in amperes (A).-carrying conductor perpendicular to the field. Measured in teslaThe SI unit of magnetic flux density. One tesla is the flux density when a force of 1 N acts on a 1 m conductor carrying 1 A perpendicular to the field. (T). (T)
$L$: length of conductor in the field (m)
$v$: velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹. of conductor perpendicular to the field (m s⁻¹)

Derivation from Faraday's law

In a small time Δt, the wire moves a distance vΔt.
The area swept out by the wire is:

\Delta A = L \times v\Delta t

The change in flux through the circuit is:

\begin{aligned} \Delta\Phi &= B \times \Delta A \\ &= BLv\Delta t \end{aligned}

Applying Faraday's lawThe magnitude of the induced EMF is proportional to the rate of change of magnetic flux linkageThe product of magnetic flux and the number of turns of a coil. Measured in weberThe SI unit of magnetic flux. One weber is the flux through an area of 1 m² when the magnetic flux density is 1 T perpendicular to the area.-turns (Wb turns).. (for a single turn, N = 1):

\begin{aligned} \varepsilon &= \frac{\Delta\Phi}{\Delta t} \\ &= \frac{BLv\Delta t}{\Delta t} \\ &= BLv \end{aligned}

Wire moving through a uniform field

A straight conductor of length L moving horizontally at velocity v through a uniform magnetic field B directed into the page. The area swept out (shaded) is L × vΔt. An EMF is induced between the ends of the wire.

Worked Example

An aircraft with a wingspan of 35 m flies horizontally at 280 m s⁻¹ through the Earth's magnetic field. The vertical component of the Earth's field is 4.5 × 10⁻⁵ T. Calculate the EMF induced between the wingtips.

Show Solution

List known values

Wingspan (conductor length): $L = 35 \text{ m}$
Speed: $v = 280 \text{ m s}^{-1}$
Vertical component of $B$: $B = 4.5 \times 10^{-5} \text{ T}$

Choose the equation

The wings cut through the vertical component of the field as the aircraft flies horizontally. Use:

$$\varepsilon = BLv$$

Substitute values

$$\varepsilon = (4.5 \times 10^{-5}) \times 35 \times 280$$

Evaluate

$$\varepsilon = 4.5 \times 10^{-5} \times 9800 = 0.44 \text{ V}$$

Answer

$\varepsilon = 0.44$ V

Common Mistake MEDIUM

Students often: Using the total Earth's magnetic field instead of the component perpendicular to the motion.
Instead: Only the component of B perpendicular to both the velocity and the length of the conductor contributes. For horizontal flight, use the vertical component of the Earth's field.