What do we mean in physics with the magnetic torque on a magnetic dipole? State

Question

What do we mean in physics with the magnetic torque on a magnetic dipole? State the equation for the magnetic torque on a magnetic dipole using magnetic dipole moment for a cirular current loop of area A. Draw a sketch supporting your reasoning.

Why do we use two different laws to determine the magnetic field of current carrying wire (or charges moving in wires) ? What are the names and equations for those two laws and what is the difference between those applying them to real world applications?

Explanation / Answer

magnetic torque, T = mue*B*sin(theta)

mue --- > magnetic dipole moment
B ----> magnetic field
theta --> angle between mue and B

Depending on distribution of current we use two laws to find magnetic field produced by a current carrying wire.

Biot-Savart law :

magnetic field due to a small current carrying segment,

dB = (mue/(4*pi))*(I*dL/r^2)

Ampere's law :

The line integral of magnetic field around a closed path is equal to mue times of the current enclosed by the loop.

integral B.dL = mue*I_enclosed

When the current distributed symetrically we use Ampere's law

and

When the current is not distributed symetrically we use Biot-Savart's law

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