A loop of wire is placed in a uniform magnetic field that is perpendicular to the plane of the loop. The strength of the magnetic field is 0.6 tesla. The area of the loop begins to... A loop of wire is placed in a uniform magnetic field that is perpendicular to the plane of the loop. The strength of the magnetic field is 0.6 tesla. The area of the loop begins to shrink at a constant rate of 0.8 m² per second. What is the magnitude of EMF induced in the loop while it is shrinking?
Understand the Problem
The question is asking to calculate the magnitude of the electromotive force (EMF) induced in a loop of wire as its area changes while under a magnetic field. The problem provides the strength of the magnetic field and the rate at which the area of the loop is shrinking, thus we can use Faraday's law of electromagnetic induction to solve it.
Answer
The magnitude of the induced EMF is given by $|\mathcal{E}| = |B \cdot \frac{dA}{dt}|$.
Answer for screen readers
The induced EMF is given by:
$$ \mathcal{E} = -B \cdot \frac{dA}{dt} $$
where $B$ is the magnetic field strength and $\frac{dA}{dt}$ is the rate of area change.
Steps to Solve
- Identify the relevant formula
We will use Faraday's law of electromagnetic induction, which states that the induced electromotive force (EMF) in a closed loop is equal to the negative rate of change of magnetic flux through the loop. The formula is given by:
$$ \mathcal{E} = -\frac{d\Phi_B}{dt} $$
where $\mathcal{E}$ is the induced EMF and $\Phi_B$ is the magnetic flux.
- Determine magnetic flux
Magnetic flux $\Phi_B$ is calculated using the equation:
$$ \Phi_B = B \cdot A $$
where $B$ is the magnetic field strength and $A$ is the area of the loop. Since the area is changing with time, we need to find the rate of change of magnetic flux.
- Express the rate of change of magnetic flux
Taking the derivative of magnetic flux with respect to time, we have:
$$ \frac{d\Phi_B}{dt} = B \cdot \frac{dA}{dt} $$
- Substitute into the EMF formula
Substituting the expression for the rate of change of magnetic flux back into Faraday's law gives us the induced EMF:
$$ \mathcal{E} = -B \cdot \frac{dA}{dt} $$
- Input the values provided
Now, replace $B$ with the strength of the magnetic field and $\frac{dA}{dt}$ with the rate at which the area is changing (make sure the values are in consistent units).
- Calculate the induced EMF
Perform the multiplication to find the magnitude of the induced EMF.
The induced EMF is given by:
$$ \mathcal{E} = -B \cdot \frac{dA}{dt} $$
where $B$ is the magnetic field strength and $\frac{dA}{dt}$ is the rate of area change.
More Information
The induced EMF can be interpreted as the energy per unit charge created by the changing magnetic field due to the shrinking area of the loop. This phenomenon is fundamental to many electrical devices, such as generators.
Tips
- Forgetting to take the absolute value when calculating the magnitude of the EMF.
- Not ensuring units are consistent (e.g., if $B$ is in Tesla, ensure area is in square meters).
- Misinterpreting the rate of area change; ensure it's negative if the area is decreasing.
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