To determine the magnitude and polarity of the electromotive force (E.M.F.) induced in a conducting rod rotating in a magnetic field, we can use Faraday's law of electromagnetic induction. This law states that the induced E.M.F. in a closed loop is equal to the rate of change of magnetic flux through the loop. In this case, we can treat the rotating rod as a segment of a loop that is cutting through the magnetic field.
Understanding the Setup
We have a conducting rod that is:
- Length: 1 cm (which is 0.01 m)
- Rotational Speed: 2 revolutions per second
- Magnetic Field Strength: 0.25 T (tesla), directed vertically upwards
Calculating the E.M.F.
The formula for the induced E.M.F. (ε) in a rotating rod in a magnetic field is given by:
ε = B * L * v
Where:
- B: Magnetic field strength (0.25 T)
- L: Length of the rod (0.01 m)
- v: Linear velocity of the rod's end, which can be calculated from the angular velocity.
Finding Linear Velocity
The linear velocity (v) at the end of the rod can be calculated using the formula:
v = r * ω
Where:
- r: Radius (length of the rod, 0.01 m)
- ω: Angular velocity in radians per second. Since the rod makes 2 revolutions per second, we convert this to radians:
ω = 2 * π * revolutions per second = 2 * π * 2 = 4π rad/s
Now, substituting the values:
v = 0.01 m * 4π rad/s ≈ 0.12566 m/s
Calculating the Induced E.M.F.
Now we can substitute the values into the E.M.F. formula:
ε = B * L * v = 0.25 T * 0.01 m * 0.12566 m/s
ε ≈ 0.00031415 V or approximately 0.314 mV.
Determining Polarity
The polarity of the induced E.M.F. can be determined using the right-hand rule. If you point your thumb in the direction of the rod's rotation (clockwise) and your fingers in the direction of the magnetic field (upwards), your palm will face the direction of the induced current. In this case, the induced current will flow in a direction that creates a magnetic field opposing the change, which means it will flow in a direction that is opposite to the magnetic field. Thus, the end of the rod that is rotating downwards will be positive, and the end that is rotating upwards will be negative.
Final Summary
In summary, the magnitude of the induced E.M.F. in the conducting rod is approximately 0.314 mV, with the polarity being such that the end of the rod rotating downwards is positive and the end rotating upwards is negative. This phenomenon illustrates the principles of electromagnetic induction effectively.
