A proton and an electron have the same kinetic energy upon entering a region of constant magnetic field. What is the ratio of the radii of their circular path?
To determine the ratio of the radii of the circular paths of a proton and an electron in a constant magnetic field, we need to apply the principles of electromagnetism.
In the presence of a magnetic field, a charged particle moves in a circular path due to the Lorentz force acting on it. The magnitude of this force is given by the equation:
F = qvB
Where:
- F is the force acting on the particle,
- q is the charge of the particle,
- v is the velocity of the particle, and
- B is the magnetic field strength.
Since the kinetic energy of a particle is given by:
KE = (1/2)mv^2
Where:
- KE is the kinetic energy of the particle, and
- m is the mass of the particle,
We can equate the magnitude of the Lorentz force and the centripetal force (required for circular motion) to relate the radius of the circular path with the kinetic energy.
mv^2/r = qvB
Canceling the v from both sides and rearranging the equation, we get:
r = (mv) / (qB)
Now, for the proton and the electron, we know that they have opposite charges, with the electron having a charge of -e and the proton having a charge of +e, where e is the elementary charge.
So, the ratio of the radii (Rproton / Relectron) is given by:
Rproton / Relectron = (mproton * vproton) / (melectron * velectron)
To continue, we need to consider the relation between the mass and charge of a proton and an electron. The mass of a proton (mproton) is approximately 1,836 times greater than the mass of an electron (melectron), making the mass of an electron significantly smaller.
Therefore, the ratio of masses (mproton / melectron) will be approximately 1,836.
Now, let's consider the velocities of the proton and electron. Since the kinetic energy of the two particles is given as equal in the question, we can equate their kinetic energies:
(1/2)mproton * vproton^2 = (1/2)melectron * velectron^2
Canceling the (1/2) factor and rearranging the equation, we find:
(mproton * vproton^2) / (melectron * velectron^2) = 1
Since the mass ratio (mproton / melectron) is approximately 1,836, we can rewrite the equation as:
(mproton / melectron) * (vproton / velectron)^2 = 1
Substituting the mass ratio, we get:
(1836) * (vproton / velectron)^2 = 1
Rearranging the equation to isolate the velocity ratio (vproton / velectron), we find:
(vproton / velectron)^2 = 1 / 1836
Taking the square root of both sides, we have:
vproton / velectron = 1 / sqrt(1836)
Now, substituting all our findings into the ratio of radii equation, we have:
Rproton / Relectron = (mproton * vproton) / (melectron * velectron)
= (1836) * (1 / sqrt(1836))
= sqrt(1836) / sqrt(1836)
= 1
Therefore, the ratio of the radii of the circular paths of a proton and an electron in a constant magnetic field is 1.