Magnetic Effects of Current and Magnetism – JEE Mains Physics

1. Biot-Savart Law and Its Application to Current Carrying Circular Loop

Biot-Savart law gives the magnetic field due to a small current element in a conductor. The magnetic field (dB) at a point due to a current element (Idl) is given by the equation:
```
dB = (μ₀/4π) * (Idl × r) / r²
```
where μ₀ is the permeability of free space, Idl is the current element, r is the distance vector, and r² is the square of the distance from the current element to the point of observation.
For a circular loop, the magnetic field at the center is given by:
```
B = (μ₀I) / (2R)
```
where I is the current and R is the radius of the loop.

Ampere's law relates the magnetic field around a current-carrying conductor to the current passing through it. It is mathematically expressed as:
```
∮B • dl = μ₀I_enclosed
```
where B is the magnetic field, dl is an infinitesimal length element, and I_enclosed is the total current enclosed by the path.
For an infinitely long straight wire, the magnetic field at a distance (r) from the wire is given by:
```
B = (μ₀I) / (2πr)
```
For a solenoid, the magnetic field is:
```
B = μ₀nI
```
where n is the number of turns per unit length and I is the current.

The force on a moving charge q in a uniform magnetic field B with velocity v is given by:
```
F = q(v × B)
```
The force is perpendicular to both the velocity and the magnetic field.
The force on a moving charge in an electric field E is given by:
```
F = qE
```
The force is in the direction of the electric field.

The force on a current-carrying conductor of length L in a uniform magnetic field B is given by:
```
F = ILB sin(θ)
```
where I is the current, L is the length of the conductor, B is the magnetic field, and θ is the angle between the magnetic field and the current.

The force per unit length between two parallel current-carrying conductors is given by:
```
F/L = (μ₀I₁I₂) / (2πr)
```
where I₁ and I₂ are the currents, r is the distance between the conductors, and μ₀ is the permeability of free space.
This force is attractive if the currents flow in the same direction and repulsive if they flow in opposite directions. The definition of the ampere is based on this interaction.

The torque (τ) experienced by a current loop of area A carrying current I in a uniform magnetic field B is given by:
```
τ = IAB sin(θ)
```
where θ is the angle between the normal to the loop and the magnetic field.
The moving coil galvanometer, which works on the principle of torque on a current loop in a magnetic field, can be converted into an ammeter or voltmeter by appropriately shunting the galvanometer.

A current loop can be treated as a magnetic dipole with a magnetic dipole moment (μ) given by:
```
μ = I A
```
where A is the area of the loop and I is the current flowing through it.

A bar magnet can be considered as a solenoid with a magnetic dipole moment. It produces a magnetic field similar to that of a solenoid.

Magnetic field lines represent the direction of the magnetic field. They form closed loops and never intersect.
The magnetic field due to a bar magnet at a point along its axis is given by:
```
B = (μ₀/4π) * (2M) / r³
```
where M is the magnetic moment and r is the distance from the center of the magnet.
The magnetic field at a point perpendicular to the axis of the bar magnet is given by:
```
B = (μ₀/4π) * M / r³
```

The torque (τ) on a magnetic dipole in a uniform magnetic field B is given by:
```
τ = μB sin(θ)
```
where θ is the angle between the magnetic moment and the magnetic field.

Paramagnetic substances: Substances that are weakly attracted to a magnetic field (e.g., aluminum).
Diamagnetic substances: Substances that are weakly repelled by a magnetic field (e.g., copper, water).
Ferromagnetic substances: Substances that are strongly attracted to a magnetic field and can be magnetized (e.g., iron, nickel, cobalt).

The magnetic properties of materials change with temperature. For ferromagnetic materials, as the temperature increases, the magnetization decreases, and at a certain temperature (Curie temperature), they lose their ferromagnetic properties and become paramagnetic.