Magnetic force

The force experienced by a moving charge in the presence of a magnetic field, which depends on charge q, velocity v and magnetic field B, and which is opposite in direction on a negative charge compared to a positive charge, is called the magnetic force.

When both electric and magnetic fields act on a charge, the total force is called the Lorentz force.

Vector Form: \[\vec F\] = q(\[\vec v\] × \[\vec B\])

Magnitude Form: F = qv B sin θ

Where:

• q = charge on the particle
• v = speed of the particle
• B = magnetic field strength
• θ = angle between \[\vec v\] and \[\vec B\]

\[\vec{E}=\frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}\hat{r}\]

Maximum magnetic force (when v ⊥ B): F_max = qv B

\[\vec F\] = q(\[\vec E\] + \[\vec v\] × \[\vec B\])

If we stretch the index finger, middle finger and thumb of the left hand mutually perpendicular to each other such that the index finger points along the direction of the magnetic field and the middle finger along the direction of current (moving charge), then the thumb represents the direction of the force F experienced by the moving charge.

Right-Hand Rule

For a positive charge, point the fingers in the direction of velocity \[\vec v\], then curl them toward the magnetic field \[\vec B\]; the thumb gives the direction of the force.

For a negative charge, the force is in the opposite direction.

Fleming's Left-Hand Rule

When the thumb, forefinger, and middle finger of the left hand are stretched mutually perpendicular to each other:

• The forefinger shows the magnetic field
• The middle finger shows current
• The thumb shows force or motion

SI Unit of Magnetic Field

The SI unit of magnetic field is tesla (T).

A magnetic field of 1 tesla is defined as the field in which a charge of 1 coulomb moving with speed 1 m/s at right angles to the field experiences a force of 1 newton.

CGS Unit

• 1 tesla = 10,000 gauss

From experiments, the magnetic force on a moving charge is found to be:

• directly proportional to charge q
• directly proportional to speed v
• directly proportional to field strength B
• directly proportional to sin⁡θ

Hence,

Combining all,

The vector form is:

Problem: A straight wire of mass 200 g and length 1.5 m carries a current of 2 A. It is suspended in mid-air by a uniform horizontal magnetic field B. Find B.

Concept used: Force on a current-carrying conductor: F = BIl

Solution:
For the wire to float, the upward magnetic force must equal the downward gravitational force:

Exam takeaway: Only the ratio m/l (mass per unit length) matters here, not the individual values.

Problem: Magnetic field B is along the +y axis. A charged particle moves along the +x axis. Find the direction of force for (a) an electron, (b) a proton.

Concept used: Vector cross product: \[\vec F\] = q(\[\vec v\] × \[\vec B\])

Solution:
Using the right-hand rule:

• For a proton (positive charge): force is along +z axis
• For an electron (negative charge): force is along the −z axis (direction reverses for negative charge)

Exam takeaway: Always apply the right-hand rule first for a positive charge, then reverse for a negative charge.

• Motion of charged particles in a cyclotron depends on the magnetic force.
• Electron beams in older CRT displays are controlled using electric and magnetic fields.
• The circular motion of charged particles in many instruments is produced by the magnetic force acting as a centripetal force.

Analogy

Magnetic force acts like a sideways push that bends the path of a moving charge without pushing it forward.

Conditions for Magnetic Force

• The charge must be moving.
• A magnetic field must be present.
• The force depends on the angle between the motion and the field.

Nature of the Force

• Magnetic force changes the direction of motion, not the speed, when only the magnetic field acts.
• Therefore, the magnetic force does no work on the charged particle.