define magnetic flux density as the force acting per unit current per unit length on a wire placed at right- angles to the magnetic field

Force on a Current‑Carrying Conductor

Learning Objective

Define magnetic flux density (\$B\$) as the force acting per unit current per unit length on a straight wire placed at right angles to a uniform magnetic field.

Key Concepts

• Magnetic field (\$\mathbf{B}\$) – a vector field that exerts a force on moving charges.
• Current (\$I\$) – the rate of flow of charge through the conductor.
• Length of wire in the field (\$L\$) – the portion of the conductor that experiences the magnetic field.
• Force on the conductor (\$\mathbf{F}\$) – given by the Lorentz force law for a current element.

Derivation of the Formula

Consider a straight conductor of length \$L\$ carrying a current \$I\$ placed perpendicular to a uniform magnetic field \$\mathbf{B}\$. The differential force on an infinitesimal element \$d\mathbf{l}\$ is

\$d\mathbf{F}=I\,d\mathbf{l}\times\mathbf{B}\$

When \$\mathbf{l}\$ is at right angles to \$\mathbf{B}\$, the cross‑product reduces to a simple multiplication of magnitudes:

\$F = I\,L\,B\$

Rearranging gives the definition of magnetic flux density:

\$B = \frac{F}{I\,L}\$

Units and Dimensions

Direction of the Force

The direction of \$\mathbf{F}\$ is given by Fleming’s Left‑Hand Rule:

1. First finger – direction of the magnetic field \$\mathbf{B}\$ (from north to south).
2. Second finger – direction of the current \$I\$ (conventional current, positive to negative).
3. Thumb – direction of the force \$\mathbf{F}\$ on the conductor.

Worked Example

Calculate the magnetic flux density if a 0.25 m long wire carrying a current of 3.0 A experiences a force of 1.2 N when placed perpendicular to the field.

1. Write the formula: \$B = \dfrac{F}{I\,L}\$.
2. Substitute the values: \$B = \dfrac{1.2\ \text{N}}{3.0\ \text{A}\times0.25\ \text{m}}\$.
3. Calculate: \$B = \dfrac{1.2}{0.75} = 1.6\ \text{T}\$.

Thus the magnetic flux density is \$1.6\ \text{tesla}\$.

Common Misconceptions

• “Magnetic field is a force.” – The field is a property of space; the force arises only when a charge or current interacts with the field.
• “The formula works for any angle.” – The simple form \$F = I L B\$ applies only when the wire is perpendicular to \$\mathbf{B}\$. For an angle \$\theta\$, \$F = I L B \sin\theta\$.
• “Current direction is the same as electron flow.” – Conventional current is defined opposite to the direction of electron motion.

Summary

Magnetic flux density \$B\$ quantifies how strong a magnetic field is at a point. It is defined by the relationship

\$B = \frac{F}{I\,L}\$

where \$F\$ is the force on a straight conductor of length \$L\$ carrying a current \$I\$, with the conductor oriented at right angles to the field. The SI unit is the tesla (T), equivalent to N·A⁻¹·m⁻¹. The direction of the force follows Fleming’s left‑hand rule.