The induced current is the highest when the direction of motion of the coil is:
at right angles to the magnetic field
Understanding Induced Current and Motion in a Magnetic Field
When a conductor moves in a magnetic field, an electromotive force (EMF) is induced across the conductor. If the conductor is part of a closed circuit, this induced EMF drives an induced current. This phenomenon is known as electromagnetic induction, and it is governed by Faraday's Law.
Factors Affecting Induced EMF and Current
The magnitude of the induced EMF depends on several factors, including the strength of the magnetic field, the length of the conductor, and the speed at which the conductor moves. Crucially, the direction of motion relative to the magnetic field also plays a significant role.
For a straight conductor of length \(L\) moving with velocity \(v\) in a uniform magnetic field \(B\), the induced EMF (\(\mathcal{E}\)) is given by the formula:
\[ \mathcal{E} = vBL\sin\theta \]Where:
- \(v\) is the speed of the conductor.
- \(B\) is the strength of the magnetic field.
- \(L\) is the length of the conductor perpendicular to both velocity and magnetic field (often just the length of the conductor).
- \(\theta\) is the angle between the direction of velocity (\(\mathbf{v}\)) and the direction of the magnetic field (\(\mathbf{B}\)).
The induced current is directly proportional to the induced EMF (according to Ohm's Law, \(I = \mathcal{E}/R\), where \(R\) is the resistance of the circuit). Therefore, the induced current will be highest when the induced EMF is highest.
Condition for Maximum Induced Current
Looking at the formula \(\mathcal{E} = vBL\sin\theta\), the induced EMF is proportional to \(\sin\theta\). The values of \(v\), \(B\), and \(L\) are constant for a given setup. To maximize the induced EMF (and thus the induced current), we need to maximize the value of \(\sin\theta\).
The maximum value of the sine function, \(\sin\theta\), is 1. This occurs when the angle \(\theta\) is \(90^\circ\) (or \(\pi/2\) radians). An angle of \(90^\circ\) means the velocity vector (\(\mathbf{v}\)) is perpendicular to the magnetic field vector (\(\mathbf{B}\)).
So, the induced EMF is maximum when the direction of motion of the conductor (or coil) is at right angles (\(90^\circ\)) to the direction of the magnetic field. Consequently, the induced current will also be maximum under this condition.
When \(\theta = 90^\circ\), \(\sin\theta = \sin 90^\circ = 1\), and the induced EMF is maximum:
\[ \mathcal{E}_{max} = vBL \]If the motion is parallel to the magnetic field (\(\theta = 0^\circ\) or \(\theta = 180^\circ\)), \(\sin\theta = 0\), and no EMF or current is induced.
Analyzing the Options
Let's examine the given options based on this understanding:
- Option 1: not at right angles to the magnetic field
If the motion is not at right angles, \(\sin\theta < 1\), so the induced EMF and current will be less than the maximum possible value. This is not the condition for the highest induced current. - Option 2: at right angles to the electricity
The concept of motion being at right angles to "electricity" is not physically relevant in electromagnetic induction. Induced current is caused by motion relative to a magnetic field, not related to the direction of pre-existing electricity. - Option 3: at right angles to the electric source
Similar to option 2, the induction phenomenon relates motion in a magnetic field to the generation of EMF, which can then drive current in a circuit. The direction of motion relative to an "electric source" is not the factor determining the maximum induced current in this context. - Option 4: at right angles to the magnetic field
As explained, when the direction of motion is at right angles (\(90^\circ\)) to the magnetic field, \(\sin\theta = 1\), resulting in the maximum induced EMF and thus the highest induced current.
Therefore, the induced current is highest when the direction of motion of the coil is at right angles to the magnetic field.
| Angle \(\theta\) between Velocity and Magnetic Field | Value of \(\sin\theta\) | Induced EMF (\(\mathcal{E} = vBL\sin\theta\)) | Induced Current (\(I \propto \mathcal{E}\)) |
|---|---|---|---|
| \(0^\circ\) (Parallel) | 0 | 0 | Zero (Minimum) |
| \(30^\circ\) | 0.5 | \(0.5 vBL\) | Intermediate |
| \(45^\circ\) | \(1/\sqrt{2} \approx 0.707\) | \(0.707 vBL\) | Intermediate |
| \(90^\circ\) (Perpendicular / Right Angle) | 1 | \(vBL\) | Highest (Maximum) |
| \(180^\circ\) (Anti-parallel) | 0 | 0 | Zero (Minimum) |
Revision Table: Electromagnetic Induction Key Points
| Concept | Description |
|---|---|
| Induced EMF | Voltage generated across a conductor due to changing magnetic flux or motion in a magnetic field. |
| Induced Current | Electric current that flows in a closed circuit due to an induced EMF. |
| Faraday's Law | States that the magnitude of the induced EMF is equal to the rate of change of magnetic flux. |
| Motional EMF | EMF induced in a conductor moving through a magnetic field, given by \(\mathcal{E} = vBL\sin\theta\). |
| Magnetic Flux (\(\Phi_B\)) | A measure of the total magnetic field lines passing through a given area, \(\Phi_B = BA\cos\phi\). |
Additional Information: Lenz's Law and Direction of Induced Current
While the magnitude of the induced current is highest when the motion is perpendicular to the magnetic field, the direction of the induced current is determined by Lenz's Law. Lenz's Law states that the direction of the induced current is such that it opposes the change in magnetic flux that produced it.
Lenz's Law is a consequence of the conservation of energy. If the induced current reinforced the change in flux, the flux change would increase, inducing more current, leading to a runaway process that creates energy out of nothing, which is impossible.
The direction can also be found using Fleming's Right-Hand Rule, particularly for motional EMF. This rule helps determine the direction of induced current when the direction of motion and magnetic field are known.
Factors like the speed of motion, the strength of the magnetic field, and the geometry of the conductor or coil influence the magnitude of the induced effect.
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