The voltage induced across a stationary conductor in an external static magnetic field
is zero
Understanding Induced Voltage in a Static Magnetic Field
The question asks about the voltage induced across a stationary conductor when it is placed in an external static magnetic field. To answer this, we need to consider the principles of electromagnetic induction, specifically Faraday's Law of Induction.
Faraday's Law of Induction states that the magnitude of the induced electromotive force (EMF), or voltage, in any closed circuit is equal to the time rate of change of the magnetic flux (ΦB) through the circuit. Mathematically, this is expressed as:
$$\mathcal{E} = - \frac{d\Phi_B}{dt}$$
Where:
- $\mathcal{E}$ is the induced voltage (EMF).
- ΦB is the magnetic flux through the circuit.
- $\frac{d\Phi_B}{dt}$ is the rate of change of magnetic flux with respect to time.
Magnetic flux ΦB is a measure of the total magnetic field passing through a given area. It is defined as ΦB = $\int_S \mathbf{B} \cdot d\mathbf{A}$, where $\mathbf{B}$ is the magnetic field vector and $d\mathbf{A}$ is an infinitesimal area vector.
In this specific scenario:
- The magnetic field is described as static. A static magnetic field is one that does not change with time ($\frac{d\mathbf{B}}{dt} = 0$).
- The conductor is described as stationary. This means its position and orientation are fixed and do not change with time.
Since the magnetic field $\mathbf{B}$ is static (constant in time) and the conductor's position and orientation are stationary (the area $d\mathbf{A}$ associated with it is constant in time), the magnetic flux ΦB passing through any area associated with the conductor will also be constant with respect to time.
If the magnetic flux ΦB is constant, its rate of change with respect to time is zero:
$$\frac{d\Phi_B}{dt} = 0$$
According to Faraday's Law, the induced voltage is directly proportional to this rate of change. Therefore, if the rate of change of magnetic flux is zero, the induced voltage is also zero.
$$\mathcal{E} = - (0) = 0$$
Let's look at the options based on this understanding:
- Option 1: "depends on the angle of the conductor with the magnetic field". This would be true if there was relative motion between the conductor and the field, or if the field was changing. However, with a stationary conductor and a static field, the induced voltage is zero regardless of the angle.
- Option 2: "increases with time". This implies a changing magnetic flux, which is not the case for a static field and a stationary conductor.
- Option 3: "is zero". This aligns perfectly with the conclusion derived from Faraday's Law for a stationary conductor in a static magnetic field.
- Option 4: "More than one of the above". Since only option 3 is correct, this is incorrect.
- Option 5: "None of the above". This is incorrect as option 3 is correct.
Therefore, the voltage induced across a stationary conductor in an external static magnetic field is zero.
Important Questions from Magnetic Field
Three infinitely long wires, each carrying equal current are placed in the xy-plane along x = 0, +d and −d. On the xy-plane, the magnetic field vanishes at
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A wire of length L is bent in the form a circular loop. And current is passed through the loop. The magnetic field induction at the centre of the loop is B. Find the current passing through the loop.
The magnetic field at the centre of a circular coil of radius r and carrying I is B. What is the magnetic field at a distance \(x = \sqrt{3}r\) from the centre, on the axis of the coil?
Two identical coils carry equal currents and have a common center, but their planes are at right angles to each other. What is the magnitude of the resultant magnetic field at the center, if field due to one coil alone is B?
