Current Carrying Wire in a Magnetic Field

A Current carrying Conductor, i.e. a current-carrying wire, will experience a Magnetic Force when encountering a Magnetic Field.

Definition §

Formula - 1 §

Force on a Current Carrying Wire in a Magnetic Field

\vec{F} = \int I \ \text{d}{\vec{l}} \times \vec{B}

##### Formula - 2 >[!formula] Force on a Straight Current Carrying Wire in a Magnetic Field > >$$ \vec{F} = BIl \sin(\theta)

$$\vec{F}\perp \vec{B}\perp \vec{I}$$

Terms §

• $\vec{F}$ = Magnetic Force (in $\text{N}$)
  • The direction is given by the Right Hand Rule
• $I$ = Current (in $\text{A}$)
• $\vec{l}$ = Length Vector (in $\text{m}$)
• $\vec{B}$ = Magnetic Field (in $\text{T}$)

Derivation §

Remember that magnetic force is experienced by a moving Charge.

Since a current is just infinitesimal flow of charge, we can use it to obtain the magnetic force. Recall the formula for current:

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

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

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

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

And then simply use the formula for magnetic force:

giving us a final formula:

$${\vec{F}}_B=q\vec{v}\times \vec{B}$$

$$\text{d}{\vec{F}}_B=\vec{v}\text{d}q\times \vec{B}$$

$$\text{d}{\vec{F}}_B=Id\vec{l}\times \vec{B}$$

$$\overrightarrow{F_B}=\int I\text{d}\vec{l}\times B$$

In the trivial case where we know that the current is constant and the line is straight, we have $\int \text{d}\vec{l}=l$

$${\vec{F}}_B=B\times Il=BIl\sin (\theta )$$

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