hi
Q: A charged particle moving with a speed 'v' enters a uniform Magnetic field 'B',perpendicular to it, now as it enters it and starts moving in a circular path of constant radius 'r' = Mv/qB (M- mass of particle), this particle has some acceleration, but MAXWELL's ELECTROAGNETIC theory says that a charged particle if it has some acceleration must lose energy in the form of radiation, so radius of the path in the above situation should gradually reduce, but it is constant WHY? (you can take charged particle as proton.)

... so radius of the path in the above situation should gradually reduce, but it is constant WHY? (you can take charged particle as proton.)

Why do you believe the radius is constant? The phenomenon you are describing is known as synchrotron radiation - to maintain constant velocity and radius of charged particle motion requires input of power to counter the energy lost through this radiation. Otherwise the partcle loses velocity and spirals in to the center.

if we balance forces we get,
Mv^2/r=qvb
which reduces to
r= Mv/qB
M,v,q,B= constant, and therefore radius constant

Why do you believe the radius is constant? The phenomenon you are describing is known as synchrotron radiation - to maintain constant velocity and radius of charged particle motion requires input of power to counter the energy lost through this radiation. Otherwise the partcle loses velocity and spirals in to the center.
http://farside.ph.utexas.edu/teaching/316/lectures/node73.html
See 167

M,v,q,B= constant, and therefore radius constant

http://farside.ph.utexas.edu/teaching/316/lectures/node73.html
see 167


I don't have access to that web page - I suspect because they restrict access to students.

But... if the particle emits synchrotron radiation then it is losing energy and its velocity is reduced. So v is not constant. I suspect that the web page you referred to describes a situation where synchrotron radiation is ignored - thus it assumes that v is constant, and you get a constant radius. The issue of synchrotron radiation becomes important only at very high speeds, when the acceleration of the particle becomes a sizable value. At relativistic speeds this presents a significant issue, but then so do some other effects that aren't covered in the equation you cited. Bottom lijne is the equation is a simplified model that is reasonably valid for particles traveling at non-relativistic speeds.