Class 10 > Unit # 16: Electromagnetism > Magnetic Force on a Moving Charge & Conductor

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Magnetic Force on Moving Charges and Conductors

(i) Force on a Moving Charge in a Magnetic Field

Let a positive charge $+q$ move with a velocity $v$ inside a uniform magnetic field of magnetic induction $B$, making an angle $\theta$ with the direction of the magnetic field. The magnitude of the magnetic force $F$ experienced by the moving charge depends upon the following factors:

• Magnitude of the charge ($q$): The greater the magnitude of the charge, the greater the force ($F \propto q$).
• Speed of the charge ($v$): The greater the speed, the greater the magnetic force ($F \propto v$).
• Magnetic field induction ($B$): The stronger the magnetic field, the greater the force ($F \propto B$).
• Angle ($\theta$): The force is proportional to the sine of the angle between velocity vector $v$ and magnetic field vector $B$ ($F \propto \sin\theta$).

Combining these factors, we get the expression for the magnetic force:

$$F = q v B \sin\theta$$

The direction of the magnetic force is given by the Right-Hand Rule (or Fleming's Left-Hand Rule).

Magnetic Field of Induction ($B$) Definition

Rearranging the force equation for a charge moving perpendicularly ($\theta = 90^\circ, \sin 90^\circ = 1$):

$$B = \frac{F}{q v}$$

"A unit magnetic field of induction is said to exist at a point where the force per unit charge moving with a velocity of $1\text{ m/s}$ in a direction perpendicular to the field is $1\text{ N}$."

The SI unit of magnetic induction $B$ is the Tesla ($\text{T}$ or $\text{N}\cdot\text{s}/(\text{C}\cdot\text{m})$).

Fig: Charged particle moving in a uniform magnetic field experiencing a perpendicular magnetic force.

(ii) Force on a Current-Carrying Conductor in a Magnetic Field

Consider a linear conductor of length $L$ carrying a steady electric current $I$, placed within a uniform magnetic field of induction $B$. Let $\theta$ be the angle between the direction of the current (length vector) and the magnetic field vector.

The total charge passing through the conductor in time $t$ is $q = I t$, and the drift velocity of the charges is $v = L / t$, which means $q v = (I t)(L / t) = I L$. Substituting this into the magnetic force formula for a moving charge gives the force on a current-carrying conductor:

$$F = I L B \sin\theta$$

Vector Form & Direction: In vector notation, the force is expressed as $\vec{F} = I (\vec{L} \times \vec{B})$. The direction of this magnetic force is mutually perpendicular to both the length vector $\vec{L}$ (current direction) and the magnetic field vector $\vec{B}$.

Fig: Conductor of length $L$ carrying current $I$ in a uniform magnetic field experiencing magnetic force $F$.

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