Magnetic Field Due To A Current Element

Self-inductance is the property of a coil by virtue of which it opposes any change in the strength of the electric current flowing through it by inducing an electromotive force (e.m.f.) in itself. It is numerically equal to the magnetic flux linked with the coil when a unit current flows through it. Its SI unit is the Henry (H).

According to the Biot-Savart law, the magnetic induction ( ) at the centre of a circular arc of wire with radius , carrying a current , and subtending an angle (in radians) at the centre is given by: where is the permeability of free space.
Magnetic Induction at the Centre of a Circular Loop:
To find the magnetic induction at the centre of a full circular loop, we consider the arc to be a complete circle.
For a complete circle, the angle subtended at the centre is radians.
Substituting into the formula for a circular arc: This is the formula for the magnetic induction at the centre of a current-carrying circular loop.

The mutual inductance ( ) between two coils is related to their self-inductances ( and ) and the coefficient of coupling ( ) by the formula: Given:
Self-inductance of the first coil,
Self-inductance of the second coil,
Coefficient of coupling,
Substitute the values into the formula: The mutual inductance between the coils is 45 mH.

The work done by the magnetic field is zero. The magnetic force on a moving charge (Lorentz force) is given by . By the definition of the cross product, the force is always perpendicular to both the velocity and the magnetic field . Since work done is , and the displacement is in the direction of velocity, the force is always perpendicular to the displacement. Therefore, the dot product is zero, and no work is done.

Given:
Galvanometer resistance,
Full scale deflection current,
(i) Conversion into an Ammeter:
To convert a galvanometer into an ammeter of range , a small resistance called a shunt ( ) is connected in parallel with it.
The required range is .
The formula for the shunt resistance is: A shunt resistance of approximately 0.201 is required.
(ii) Conversion into a Voltmeter:
To convert a galvanometer into a voltmeter of range , a large resistance ( ) is connected in series with it.
The required range is .
The formula for the series resistance is: A series resistance of 4950 is required.

For a long solenoid, the magnetic field ( ) at its centre is uniform and given by the formula: where is the permeability of free space ( ), is the current, and is the number of turns per unit length.
Given:
Length,
Number of turns,
Current,
First, calculate the number of turns per unit length ( ): Now, calculate the magnetic field : The terms cancel out: The magnetic field at the centre of the solenoid is Tesla. (Note: The diameter of the solenoid is not needed for this calculation, assuming it is a long solenoid where end effects are negligible).

Consider a conducting bar of length rotating with a constant angular velocity about a pivot at one end. The rotation occurs in a uniform magnetic field which is perpendicular to the plane of rotation.
Consider a small element of the bar of length at a distance from the pivot.
The linear velocity of this element is .
The motional e.m.f. induced in this small element is given by: Substituting : To find the total e.m.f. induced across the entire length of the bar, we integrate this expression from the pivot ( ) to the other end ( ): Since and are constant: This is the expression for the rotational e.m.f. induced in the conducting bar.

Four Uses of Van de Graaff Generator:
Nuclear Physics Research: It is used to accelerate charged particles (like protons, deuterons) to high energies. These high-energy particles are then used to bombard atomic nuclei to study nuclear reactions and structure.
Particle Accelerators: It often serves as the initial stage (injector) for larger particle accelerators.
Medical Applications: High-energy beams produced by the generator can be used to generate X-rays for radiation therapy in treating cancer.
Industrial Applications: The high-energy X-rays can be used for non-destructive testing of materials and welds, and the electron beams can be used for sterilization.
Calculation:
We need to calculate the displacement current ( ). Given:
Rate of change of electric field,
Area of plates,
Constant:
Permittivity of free space,
The displacement current is given by Maxwell's equation: where is the electric flux. For a parallel plate capacitor with a uniform field, . Since the area is constant: Substituting this into the formula for : The displacement current is 3.54 mA.