Moving Charges | Formulae and Concept REVISION in 13 min | JEE Physics by Mohit Sir (IIT KGP)

Main Ideas and Formula Revision

The video is a rapid review of magnetic fields produced by currents and the motion and forces of charged particles in magnetic fields. It emphasizes remembering formula conditions, directions, and common applications.

1. Magnetic Field from Currents

• Biot–Savart law: For a small current element (I\,d\boldsymbol{\ell}), the magnetic-field contribution at a point a distance (r) away is [ d\mathbf B=\frac{\mu_0}{4\pi}\frac{I\,d\boldsymbol{\ell}\times\hat{\mathbf r}}{r^2}. ] Use the right-hand rule to determine the direction. Fields from multiple current elements or moving charges add by superposition.

• Straight current-carrying wire: At perpendicular distance (a) from a finite wire, [ B=\frac{\mu_0 I}{4\pi a}(\sin\theta_1+\sin\theta_2), ] where the angles are measured from the perpendicular to the wire. Useful cases are:

  • Semi-infinite wire: (B=\mu_0 I/(4\pi a))
  • Infinite wire: (B=\mu_0 I/(2\pi a))

Use the right-hand thumb rule to find the field direction.

• Circular loop: On the axis of a loop of radius (R), at distance (x) from its center, [ B=\frac{\mu_0 NIR^2}{2(R^2+x^2)^{3/2}}. ] At the center, this becomes (B=\mu_0 NI/(2R)). For a circular arc subtending angle (\theta) radians at the center, [ B=\frac{\mu_0 I\theta}{4\pi R}. ]

• Long solenoid: For an ideal long solenoid, the field inside is approximately [ B=\mu_0 nI, ] where (n) is the number of turns per unit length.

• Ampère’s law: [ \oint \mathbf B\cdot d\boldsymbol{\ell}=\mu_0 I_{\text{enclosed}}. ] Choose an Ampèrian loop that suits the symmetry of the problem. Count the enclosed current with a sign according to its direction through the loop. The magnetic field at a point can be produced by all currents, even though the Ampère integral uses the net current enclosed by the chosen loop.

2. Force and Motion of a Charged Particle

• Magnetic force: [ \mathbf F=q\,\mathbf v\times\mathbf B,\qquad F=|q|vB\sin\theta. ] The force is perpendicular to the instantaneous velocity, so a magnetic field alone does no work and does not change a particle’s speed.

• Velocity perpendicular to the field: The particle moves in a circle: [ r=\frac{mv}{|q|B},\qquad T=\frac{2\pi m}{|q|B},\qquad \omega=\frac{|q|B}{m}. ] If accelerated through a potential difference (V), then (|q|V=\tfrac12 mv^2), which can be combined with the radius formula.

• Time spent in a magnetic-field region: If a particle traces an arc of angle (\theta), its time in the field is [ t=\frac{\theta}{2\pi}T. ] A semicircular path therefore corresponds to half a cyclotron period. The path’s radius and the region’s dimensions determine whether the particle exits along a semicircle or a shorter arc.

• Velocity at an angle to the field: Resolve the velocity into components:

  • (v_\perp) produces circular motion, with radius (r=mv_\perp/(|q|B)).
  • (v_\parallel) carries the particle along the field.

Together, they produce a helical path; the distance advanced along the field in one period is the helix’s pitch.

• Crossed electric and magnetic fields: If the electric and magnetic forces oppose each other, a particle passes undeflected when [ |q|E=|q|vB,\qquad v=\frac{E}{B}. ] Check the force directions using the charge’s sign and the field directions.

3. Forces and Moments Involving Current

• Force on a current-carrying wire: [ \mathbf F=I\,\mathbf L\times\mathbf B,\qquad F=ILB\sin\theta. ]

• Parallel wires: Two parallel currents in the same direction attract; currents in opposite directions repel. The force per unit length is [ \frac{F}{L}=\frac{\mu_0 I_1 I_2}{2\pi d}, ] where (d) is the distance between the wires.

• Current loop in a magnetic field: A loop’s magnetic dipole moment is [ \boldsymbol{\mu}=NIA\,\hat{\mathbf n}, ] with direction given by the right-hand rule. In a uniform field, a closed loop has zero net force but can experience torque: [ \boldsymbol{\tau}=\boldsymbol{\mu}\times\mathbf B,\qquad U=-\boldsymbol{\mu}\cdot\mathbf B. ] In a non-uniform magnetic field, the forces on different parts of the loop may not cancel, so the loop can experience a net force.

Speakers or Sources Featured

• Mohit Sir — the sole presenter, explaining the formulas and concepts. The subtitles also refer to a JEE exam question as an example.