Class 12 > Unit # 18:Magnetic Fields > Magnetic Field & Magnetic Force on Current Carrying Conductor

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Magnetic Fields - Magnetic Force on Conductors

Define Magnetic field and explain how magnetic field is produced due to current.

Magnetic Field

"A magnetic field is a region in space in which a force is experienced by a moving electric charge or a magnetic material."

Magnetic Field Due to Current

It was discovered by Oersted that when electric current passes through a conductor, a magnetic field is produced around it. This field is known as the magnetic field of induction and is denoted by the symbol “B”.

Ampere further found that when two current-carrying conductors are placed near each other, they experience a mutual magnetic force:

• If the electric currents flow in the same direction, the magnetic force between them is attractive.
• If the electric currents flow in opposite directions, the magnetic force between them is repulsive.

When electrical charges are at rest, they exert only an electrostatic force on each other. However, when charges are in motion (forming an electric current), they create both electric and magnetic fields, thereby exerting both electric and magnetic forces on one another.

Q. Derive an expression for the force on a current carrying conductor present in a magnetic field.

Factors on Which Force Acts

The magnetic force acting on a current-carrying conductor depends on several physical factors:

1. The force ($F$) is directly proportional to the length of the conductor ($L$) lying within the magnetic field:
   $F \propto L$
2. The force is directly proportional to the current ($I$) passing through the conductor:
   $F \propto I$
3. The force is directly proportional to the strength of the external magnetic field ($B$):
   $F \propto B$

Derivation of the Force Expression

Combining these proportionalities together, we obtain:

$F \propto I \cdot L \cdot B \implies F = k I L B$

Here, $k$ is the proportionality constant, and in the SI unit system, its value is equal to $1$. Thus, the force is expressed as:

$F = I L B \quad \text{--- (1)}$

If the magnetic field vector is not perpendicular to the wire but makes an angle $\theta$ with the length of the conductor, Equation (1) is modified using the vector cross product components:

$F = I L B \sin\theta$

Special Cases for Magnitude of Force

• Maximum Force: The deflecting force reaches its maximum value when the angle between the length of the conductor and the magnetic field is $\theta = 90^\circ$ ($\sin 90^\circ = 1$):
  $F_{\text{max}} = I L B$
• Minimum Force: The deflecting force is minimized when $\theta = 0^\circ$ or $180^\circ$ ($\sin 0^\circ = 0$), signifying that the conductor is aligned parallel to the magnetic field:
  $F_{\text{min}} = 0$

Direction of Force

Since magnetic force is a vector quantity, its spatial direction is determined using Fleming's left-hand rule, which establishes the orthogonal relationship between the force, current, and magnetic field directions.

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