Chapter 5.1 and 5.2

Magnetic Forces

Magnetic Flux Density: $\vec B$

• Fire a test charge $+q$ with velocity $\vec u$ through a point $P$
• If a force $\vec F_m$ acts on moving $+q$, then a magnetic field $\vec B$ in present at point P such that $F_m = quB \sin \theta$

$$ F_m = q \vec u \times \vec B \text{ [T]} $$

• Direction of $F_m$ is given by right hand rule

• If both $\vec E$ and $\vec B$ are present then electromagnetic force is Lorentz force:

  $$ \vec F = q(\vec E + \vec u \times \vec B) $$

Difference between $F_e$ and $F_m$

• $F_e$ and $E$ have same directions
• $F_m$ and $B$ are perpendicular
• $F_e$ acts on charged particle in motion or stationary
• $F_m$ acts only on $q$ in motion
• $F_e$ does work on moving $q$
• $F_m$ foes not do work on moving $q$, only changes direction
  • Because the magnetic force $F_m$ is always perpendicular to $u$, $F_m \cdot u = 0$
  • Since no work is done, a magnetic field cannot change the kinetic energy of a charged particle,
  • The magnetic field can change the direction of motion of a charged particle, but not its speed

Magnetic force on a current-carrying conductor (wire)

The vector sum of the infinitesimal vectors $d l$ over a closed path = 0. In other words, the total magnetic force on any closed current loop in a uniform magnetic field is zero

• $\vec B$ exerts a sideways force on a moving charge
• Current $I$ is moving charges