An electron and a proton execute uniform circular motion in the presence of a uniform magnetic field.
If the radius of the circle is the same for both particles, which of the following statements is true?
a. Both particles move in the same direction; the electron moves faster than the proton.
b. Both particles move in the same direction; the proton moves faster than the electron.
c. The electron and the proton move in opposite directions but they move with the same speed.
d. The electron and the proton move in opposite directions; the electron moves faster than the proton.
e. The electron and the proton move in opposite directions; the proton moves faster than the electron.
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To determine the correct statement, we can use the formula for the force experienced by a charged particle in a magnetic field.
The force experienced by a charged particle moving in a magnetic field is given by the equation: F = qvBsinθ
Where:
F = Force experienced by the particle
q = Charge of the particle
v = Velocity of the particle
B = Magnetic field strength
θ = Angle between the velocity vector and the magnetic field vector
In the case of circular motion, the force acting on the particle is the centripetal force, which is given by the equation: F = mv^2/r
Where:
m = Mass of the particle
v = Velocity of the particle
r = Radius of the circular path
Since both the electron and the proton have the same radius, we can equate the centripetal force to the magnetic force:
mv^2/r = qvBsinθ
Canceling out the velocity, we get:
mv/r = qBsinθ
Now, let's analyze the given options:
a. Both particles move in the same direction; the electron moves faster than the proton.
This statement is not correct because if the electron were moving faster, it would have a greater velocity, which would result in a larger force acting on it. This contradicts the equation we derived where the forces are equal.
b. Both particles move in the same direction; the proton moves faster than the electron.
This statement is also not correct because if the proton were moving faster, it would have a greater velocity, which would result in a larger force acting on it. This contradicts the equation we derived where the forces are equal.
c. The electron and the proton move in opposite directions, but they move with the same speed.
This statement is not correct because in order for the forces to be equal, the charges of the particles would need to be equal, which is not the case. Electrons have a charge of -1 and protons have a charge of +1.
d. The electron and the proton move in opposite directions; the electron moves faster than the proton.
This statement is correct. According to our equation, since the forces are equal and the electron has a smaller mass and a larger charge than the proton, it must have a greater velocity to balance out the equation.
e. The electron and the proton move in opposite directions; the proton moves faster than the electron.
This statement is not correct because, as explained above, the electron must move faster than the proton to balance out the equation.
Therefore, the correct statement is d. The electron and the proton move in opposite directions; the electron moves faster than the proton.