How can a magnetic field accelerate particles if it cannot do work? A varying magnetic field can accelerate charge particles, but it is said that a magnetic field can't do any work so it should not be able to speed up charged particles, right? How is this apparent contradiction resolved?
 A: A varying magnetic field generates an electric field, and an electric field can do work on a particle.  This is called Faraday's law of induction:
$$\nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}$$
The full Lorentz force equation is 
$$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$$
So for example, if the magnetic field is increasing in the $\hat{z}$ direction, such that
$$\vec{B} = b t \hat{z}$$
and
$$\frac{\partial \vec{B}}{\partial t} = b \hat{z}$$
then the electric field is determined by
$$\nabla \times \vec{E} = - b \hat{z}$$
Thus the electric field is not zero, so work can be done on a charged particle as a result of a changing magnetic field.
A: The force on a charged particle is called the Lorentz force, and it is give by:
$$ {\bf F} = q({\bf v} \times {\bf B}) $$
where the $\times$ symbols means a cross product. This means the force ${\bf F}$ is always at right angles to the direction of motion ${\bf v}$, and therefore the work done on the charged particle is zero. The Lorentz force can accelerate the particle by changing the direction of its velocity but it cannot change the magnitude of the velocity.
A: The answers you get refer the phenomenon of deflection of charged particles in a magnetic field to a quantitative description, derived from experiments. The mechanism, how this happens, stays hidden.
The electron has three well known properties, its electric charge, its magnetic dipole moment and its intrinsic spin. All three are constant quantities. And to prevent contradiction about the reality of this intrinsic spin, it was shown in the Einstein-de-Haas experiment, that this spin really has to do with a rotation of the electron.
It has to be stated, that the magnetic dipole moment and the intrinsic spin in the electron are aligned. This is a very important fact for the following explanation.
Being under the influence of a magnetic field, the electron's magnetic dipole moment get aligned. If the electron is not moving or if the electron moved parallel to the magnetic field that is it, nothing more happens.
But if the electron moves non parallel to the external magnetic field there came in the game the electron intrinsic spin. Due to the torque induced precession (gyroscopic effect) a rotating body tries to resist its deflection. One can feel this by deflecting a rotating wheel from a bicycle. The magnetic field align the magnetic dipole moment and this time the intrinsic spin too. The spin resists against the deflection and emitting photons go back in the direction of his previous state. This repeats many times until the electron comes to rest.
What we see is an electron's movement on a spiral path and the emission of photons.
A: Here, the magnetic field doesn't change the speed of the particle, but it changes the direction of the particle hence accelerating the particle.
No work is done by the field because the force exerted by the magnetic field is perpendicular to the displacement of the particle and word done is their dot product and hence their dot product is zero because dot product of perpendicular vectors is zero
