Torque on a loop in which current is induced
Torque on a Loop in Which Current is Induced
Electromagnetic induction is a fundamental principle in physics that describes the process by which a changing magnetic field within a loop of wire induces an electric current. When a loop of wire is placed in a magnetic field and the amount of magnetic flux through the loop changes, an electromotive force (EMF) is induced, which can drive a current if the loop is part of a closed circuit.
Torque on a Current-Carrying Loop
When a current-carrying loop is placed in a magnetic field, it experiences a torque due to the interaction between the magnetic field and the magnetic moment generated by the current in the loop. This torque tends to rotate the loop so that its plane becomes perpendicular to the magnetic field lines.
The torque ((\vec{\tau})) on a current-carrying loop in a uniform magnetic field ((\vec{B})) is given by the cross product of the magnetic moment ((\vec{\mu})) of the loop and the magnetic field:
[ \vec{\tau} = \vec{\mu} \times \vec{B} ]
The magnitude of the torque can be expressed as:
[ \tau = \mu B \sin(\theta) ]
where:
- (\mu = NIA) is the magnetic moment of the loop, with (N) being the number of turns, (I) the current, and (A) the area of the loop.
- (B) is the magnitude of the magnetic field.
- (\theta) is the angle between the normal to the plane of the loop and the magnetic field.
Induced Current in a Loop
Faraday's law of electromagnetic induction states that the induced EMF in a loop is equal to the negative rate of change of magnetic flux through the loop:
[ \text{EMF} = -\frac{d\Phi_B}{dt} ]
where (\Phi_B = B \cdot A \cdot \cos(\theta)) is the magnetic flux through the loop.
If the loop has a resistance (R), Ohm's law relates the induced EMF to the induced current (I):
[ I = \frac{\text{EMF}}{R} ]
Torque on a Loop with Induced Current
When a current is induced in a loop by a changing magnetic field, the loop will experience a torque similar to the one experienced by a current-carrying loop. The torque on the loop with induced current can be calculated using the same formula as above, with the induced current (I) replacing the constant current.
Differences and Important Points
| Aspect | Current-Carrying Loop | Loop with Induced Current |
|---|---|---|
| Source of Current | External power supply | Changing magnetic field |
| Current | Constant (DC) or varying (AC) | Induced by change in flux |
| Torque Calculation | Use constant current in formula | Use induced current in formula |
| Magnetic Moment | (\mu = NIA) | (\mu = NI_{\text{induced}}A) |
| Induced EMF | Not applicable | (\text{EMF} = -\frac{d\Phi_B}{dt}) |
| Resistance Consideration | Not directly involved | Affects induced current via Ohm's law |
Example: Calculating Torque on a Loop with Induced Current
Consider a rectangular loop with (N) turns, area (A), and resistance (R), placed in a magnetic field (B) that is changing with time. The loop's plane is perpendicular to the magnetic field. The rate of change of the magnetic field is given by (\frac{dB}{dt}).
The induced EMF in the loop is:
[ \text{EMF} = -N \frac{d\Phi_B}{dt} = -NA \frac{dB}{dt} ]
The induced current in the loop is:
[ I_{\text{induced}} = \frac{\text{EMF}}{R} = -\frac{NA}{R} \frac{dB}{dt} ]
The torque on the loop is:
[ \tau = NIA \cdot B \sin(\theta) ]
Since the loop's plane is perpendicular to the magnetic field, (\theta = 90^\circ) and (\sin(\theta) = 1). Therefore, the torque is:
[ \tau = N \left(-\frac{NA}{R} \frac{dB}{dt}\right) A \cdot B ]
[ \tau = -\frac{N^2 A^2 B}{R} \frac{dB}{dt} ]
This torque will cause the loop to rotate, and the direction of the torque can be determined using the right-hand rule for cross products.
Understanding the torque on a loop with induced current is crucial in the design and operation of electric motors and generators, where the principle of electromagnetic induction is applied to convert electrical energy into mechanical energy, and vice versa.