To solve the problem of finding the electromotive force (EMF) induced across a rod placed in a circular magnetic field that is increasing at a constant rate, we can apply Faraday's law of electromagnetic induction. Let's break this down step by step.
Understanding the Setup
We have a circular magnetic field with a radius \( R \) that is increasing at a constant rate, denoted as \( \alpha \). The rod, which has a length of \( 2R \), is tangential to this magnetic field and lies in the same plane as the field. The key here is to understand how the changing magnetic field induces an EMF in the rod.
Applying Faraday's Law
Faraday's law states that the induced EMF (\( \mathcal{E} \)) in a closed loop is equal to the negative rate of change of magnetic flux (\( \Phi_B \)) through that loop. Mathematically, it is expressed as:
\( \mathcal{E} = -\frac{d\Phi_B}{dt} \)
Calculating Magnetic Flux
The magnetic flux through the area enclosed by the rod can be calculated using the formula:
\( \Phi_B = B \cdot A \)
Where \( B \) is the magnetic field strength and \( A \) is the area. Since the magnetic field is circular and increasing, we can express the magnetic field strength \( B \) as a function of time:
\( B(t) = B_0 + \alpha t \)
Here, \( B_0 \) is the initial magnetic field strength at time \( t = 0 \), and \( \alpha \) is the rate of increase of the magnetic field.
Area Calculation
The area \( A \) enclosed by the rod can be considered as a sector of a circle. The angle \( \theta \) subtended by the rod at the center of the circle is \( \frac{2R}{R} = 2 \) radians (since the length of the rod is \( 2R \)). The area of this sector can be calculated as:
\( A = \frac{1}{2} R^2 \theta = \frac{1}{2} R^2 \cdot 2 = R^2 \)
Finding the Rate of Change of Flux
Now, substituting \( B(t) \) and \( A \) into the flux equation, we get:
\( \Phi_B = (B_0 + \alpha t) \cdot R^2 \)
To find the induced EMF, we differentiate the magnetic flux with respect to time:
\( \frac{d\Phi_B}{dt} = R^2 \cdot \frac{d(B_0 + \alpha t)}{dt} = R^2 \cdot \alpha \)
Calculating the Induced EMF
Now, substituting this back into Faraday's law gives us:
\( \mathcal{E} = -\frac{d\Phi_B}{dt} = -R^2 \cdot \alpha \)
Since EMF is typically expressed as a positive quantity, we can write:
\( \mathcal{E} = R^2 \cdot \alpha \)
Final Result
The induced EMF across the rod, which is tangential to the circular magnetic field that is increasing at a constant rate, is given by:
\( \mathcal{E} = R^2 \cdot \alpha \)
This result shows that the induced EMF is directly proportional to the square of the radius of the magnetic field and the rate of increase of the magnetic field. This relationship highlights the significance of both the geometry of the setup and the dynamics of the magnetic field in electromagnetic induction.
