Motional EMF: Videos & Practice Problems

Motional EMF arises from the movement of a conductor through a magnetic field, leading to induced electric charges due to the magnetic force acting on moving charges. The relationship is defined by the equation $E^m=vBl$, where Em is the electromotive force (emf), v is the velocity, B is the magnetic field strength, and l is the length of the conductor. This principle is a specific application of Faraday's law, emphasizing the importance of changing magnetic flux in circuits.

Motional EMF

Motional electromotive force (EMF) is a specific instance of Faraday's law, which states that a changing magnetic flux through a loop induces an EMF. In the case of motional EMF, this change occurs due to the motion of a conductor within a magnetic field. For example, consider a conducting rod moving through a magnetic field. As the rod moves, the charges within it experience a magnetic force, which can be determined using the equation:

$F_b=qvb\backslash sin\mathrm{\backslash theta}$

Here, \( F_b \) is the magnetic force, \( q \) is the charge, \( v \) is the velocity of the charge, \( b \) is the magnetic field strength, and \( \theta \) is the angle between the velocity and the magnetic field. In the case where the angle is 90 degrees, the sine term can be simplified, leading to a direct relationship between the electric field \( E \) and the velocity \( v \) and magnetic field \( b \):

$E=vb$

This induced electric field results in a separation of charges within the rod, creating an induced EMF, which can be expressed as:

$\mathrm{\backslash epsilon}=vbl$

In this equation, \( \epsilon \) represents the induced EMF, \( l \) is the length of the conducting rod, and \( v \) is the velocity of the rod moving through the magnetic field. This relationship shows that motional EMF is essentially an application of Faraday's law, where the changing variable is the area of the loop formed by the moving rod.

When the rod is part of a closed circuit, the induced EMF can be used to calculate the induced current using Ohm's law:

$I=\mathrm{\backslash epsilon}R$

Where \( I \) is the induced current and \( R \) is the resistance of the circuit. For example, if the rod has a length of 10 cm, a magnetic field strength of 0.2 Tesla, and moves at a velocity of 25 m/s, with a resistance of 10 milliohms, the induced current can be calculated as follows:

$I=vblR$

Substituting the values gives:

$I=\backslash frac\{(25)(0.2)(0.1)\}\{0.010\}\; =\; 0.5\; \backslash text\{\; A\}$

To find the power output in the circuit, we can use the formula:

$P=I^2R$

Substituting the induced current and resistance yields:

$P=(0.5)^2\; \backslash cdot\; 0.010\; =\; 2.5\; \backslash times\; 10^\{-3\}\; \backslash text\{\; W\}\; =\; 2.5\; \backslash text\{\; mW\}$

This demonstrates how motional EMF can be applied to practical problems involving induced currents and power in circuits.

A thin rod moves in a perpendicular, unknown magnetic field. If the length of the rod is 10 cm and the induced EMF is 1 V when it moves at 5 m/s, what is the magnitude of the magnetic field?

• A. 0.02 T
• B. 0.5 T
• C. 2 T
• D. 50 T

Answer: (C)

Forces on Loops Exiting Magnetic Fields

In this problem, we analyze a rectangular loop being pulled out of a magnetic field at a constant velocity. To determine the magnitude and direction of the induced current when the loop is halfway out of the field, we apply Faraday's law of electromagnetic induction, specifically focusing on motional electromotive force (EMF).

The induced current can be calculated using the formula:

$$ I_{\text{mot}} = \frac{VBL}{R} $$

where:

• \( I_{\text{mot}} \) is the induced current,
• \( V \) is the velocity of the loop (12 m/s),
• \( B \) is the magnetic field strength (0.5 T),
• \( L \) is the length of the loop (0.2 m), and
• \( R \) is the resistance (0.4 Ω).

Substituting the values, we find:

$$ I_{\text{mot}} = \frac{(12 \, \text{m/s})(0.5 \, \text{T})(0.2 \, \text{m})}{0.4 \, \Omega} = 3 \, \text{A} $$

Next, we determine the direction of the induced current using Lenz's law, which states that the induced current will flow in a direction that opposes the change in magnetic flux. The magnetic field is directed into the page, and as the loop exits the field, the area exposed to the magnetic field decreases, leading to a negative change in magnetic flux.

To counteract this decrease, the induced magnetic field must also point into the page. Using the right-hand rule, if you point your thumb into the page, your fingers curl in the direction of the induced current, which is clockwise. Thus, the direction of the induced current is clockwise.

For the second part of the problem, we need to find the magnitude of the external force required to maintain a constant velocity as the loop exits the magnetic field. Since the loop is moving at constant velocity, the net force acting on it must be zero, meaning the external force must balance the magnetic force acting on the loop due to the induced current.

The magnetic force \( F \) on a current-carrying conductor in a magnetic field is given by:

$$ F = I \cdot L \cdot B $$

Setting the external force equal to the magnetic force, we have:

$$ F_{\text{external}} = I_{\text{mot}} \cdot L \cdot B $$

Substituting the known values:

$$ F_{\text{external}} = (3 \, \text{A})(0.2 \, \text{m})(0.5 \, \text{T}) = 0.3 \, \text{N} $$

Thus, the external force required to keep the loop moving at a constant velocity while exiting the magnetic field is 0.3 newtons.

Here’s what students ask on this topic:

What is motional EMF and how is it related to Faraday's law?

Motional EMF arises when a conductor moves through a magnetic field, causing an induced electromotive force (EMF) due to the magnetic force acting on the moving charges. This is a specific application of Faraday's law, which states that a changing magnetic flux through a loop induces an EMF. In the case of motional EMF, the change in magnetic flux is due to the motion of the conductor. The relationship is given by the equation:

$\epsilon =vBl$

where $\epsilon$ is the EMF, $v$ is the velocity of the conductor, $B$ is the magnetic field strength, and $l$ is the length of the conductor.

How do you calculate the induced current in a circuit with motional EMF?

To calculate the induced current in a circuit with motional EMF, you first need to determine the induced EMF using the equation:

$\epsilon =vBl$

where $v$ is the velocity of the conductor, $B$ is the magnetic field strength, and $l$ is the length of the conductor. Once you have the EMF, use Ohm's law to find the induced current:

$I=\frac{\epsilon }{R}$

where $I$ is the induced current and $R$ is the resistance of the circuit.

What is the significance of the right-hand rule in determining the direction of the magnetic force in motional EMF?

The right-hand rule is crucial for determining the direction of the magnetic force in motional EMF. To apply it, point your fingers in the direction of the magnetic field (B), and your thumb in the direction of the velocity (v) of the conductor. The direction your palm faces indicates the direction of the magnetic force (F) on positive charges. This helps in understanding how charges separate within the conductor, leading to the creation of an electric field and the resulting EMF.

How does the area change in a circuit affect the induced EMF according to Faraday's law?

According to Faraday's law, the induced EMF in a circuit is proportional to the rate of change of magnetic flux through the circuit. The magnetic flux (Φ) is given by:

$\Phi =BAcos\theta$

where $B$ is the magnetic field strength, $A$ is the area of the loop, and $\theta$ is the angle between the magnetic field and the normal to the loop. If the area (A) changes, the magnetic flux changes, leading to an induced EMF. The induced EMF (ε) is given by:

$\epsilon =\frac{-d\Phi }{\mathrm{dt}}$

What is the relationship between motional EMF and the length of the conductor?

The relationship between motional EMF and the length of the conductor is direct and linear. The induced EMF (ε) is given by the equation:

$\epsilon =vBl$

where $v$ is the velocity of the conductor, $B$ is the magnetic field strength, and $l$ is the length of the conductor. This means that if the length of the conductor increases, the induced EMF also increases proportionally, assuming the velocity and magnetic field strength remain constant.