Magnetic Field From A Current

In a Current carrying wire, there exist numerous moving charges, and hence a Magnetic Field is generated. It can be calculated using the Biot-Savart Law

Magnetic Field §

Assume have an infinitely thin conducting wire, carrying a current $I$ and we want to measure the magnetic field at a point $P$. Then:

Formula §

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$$\vec{B}=\int \frac{{\mu }_0}{4\pi }\frac{I}{r^2}\hat{u}\times \text{d}\vec{l}$$

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• $\vec{B}$ = Magnetic Field (in $T$ or $N/Am$)
• $I$ = Current (in $A$)
• $\vec{l}$ = Path vector of the wire
• $\hat{u}$ = Direction vector pointing from $\vec{l}$ to $P$
• $r$ = Distance from $d\vec{l}$

Derivation §

The Current of the wire is given by

The Biot-Savart Law gives:

$$I=\frac{\text{d}q}{\text{d}t}\to \text{d}q=I\text{d}t$$

Also we know that the speed of an infinitesimal charge $\text{d}q$ is given by

$$\vec{v}=\frac{\text{d}\vec{l}}{\text{d}t}\to \text{d}t=\frac{d\vec{l}}{\vec{v}}$$

Combining the two:

$$\text{d}q=I\text{d}t=\frac{I}{\vec{v}}d\vec{l}$$

$$d\vec{B}=\frac{{\mu }_0}{4\pi }\frac{I}{r^2}\hat{u}\times d\vec{l}$$

Giving us:

$$\vec{B}=\int \frac{{\mu }_0}{4\pi }\frac{I}{r^2}\hat{u}\times \text{d}\vec{l}$$