A proton follows a spiral path througha gas in a magnetic field of .010 T , perpendicular to the plane of the spiral.In two successive loops, at point P and Q, the radii are 10.0mm and 8.5 mm,respectively.Calculate the change in Kinetic energy of the proton as it travels from P to Q.
To calculate the change in kinetic energy of the proton as it travels from point P to point Q, we can use the Lorentz force equation and the principle of conservation of energy.
The Lorentz force equation states that the force experienced by a charged particle moving in a magnetic field is given by:
F = q * v * B
Where:
F is the force experienced by the particle,
q is the charge of the particle (in this case, the charge of a proton is +1.60 x 10^-19 C),
v is the velocity of the particle, and
B is the magnetic field strength (given as 0.010 T).
In this case, the proton is moving in a spiral path, so its velocity is continuously changing. However, since we want to calculate the change in kinetic energy, we can assume that the average force acting on the proton over the distance from P to Q is equal to the net force acting on the proton.
The net force acting on the proton can be calculated by subtracting the forces at points P and Q:
Net Force = Fq - Fp
Now, let's calculate the forces at points P and Q.
At point P:
Radius (Rp) = 10.0 mm = 0.0100 m
Centripetal Force at P = m * (v^2 / Rp)
v = velocity at P (which is equal to the velocity at Q because we are assuming constant speed)
m = mass of the proton = 1.67 x 10^-27 kg
Now the centripetal force in terms of potentials
Rp.m.v^2 = q.v.B.Rp
At point Q:
Radius (Rq) = 8.5 mm = 0.0085 m
Centripetal Force at Q = m * (v^2 / Rq)
Similar to above,
Rq.m.v^2 = q.v.B.Rq
Now we can calculate the difference in the centripetal forces at points P and Q:
Net Force = m * (v^2 / Rq) - m * (v^2 / Rp)
Net Force = m * [(v^2 / Rq) - (v^2 / Rp)]
Now, to calculate the change in kinetic energy (ΔK), we can use the work-energy theorem:
ΔK = Work done by the net force = Net Force * Distance (d)
The distance traveled by the proton from point P to point Q is equal to the difference in radii:
d = Rq - Rp
Finally, we can calculate the change in kinetic energy:
ΔK = Net Force * d
Substituting the values we calculated earlier, we can find the change in kinetic energy of the proton as it travels from point P to point Q.