A proton traveling at 10^4 m/s in the positive x-direction, enters a region where there is uniform electric field of magnitude 10 v/m and a uniform magnetic field of magnitude 10^-2 T. The electric and magnetic fields are parallel and are in the z-direction. In the following statements which of the two are false
A. When the proton enters the fields, the magnitude of magnetic force on it is ten times the magnitude of electric force
B. The initial acceleration of the proton is in the y-z plane, at 84 degrees to the z-axis and at 6 degrees to the negative y axis
C. When the proton enters the fields the magnetic force on it is perpendicular to the electric field
D. The electric force on the proton is constant
E. The magnetic force on the proton is constant
F. The speed of the proton changes with time
G. The path of the proton is a uniform helix
Let's analyze each statement one by one:
A. When the proton enters the fields, the magnitude of magnetic force on it is ten times the magnitude of electric force.
This statement is True. The magnetic force on a charged particle moving through a magnetic field is given by the equation Fm = qvB, and the electric force is given by Fe = qE. Given that the magnetic field is 10^-2 T and the electric field is 10 V/m, the ratio of the magnetic force to the electric force is (qvB)/(qE) = (vB)/E = (10^4)*(10^-2)/(10) = 10. Therefore, the magnetic force is indeed ten times the magnitude of the electric force.
B. The initial acceleration of the proton is in the y-z plane, at 84 degrees to the z-axis and at 6 degrees to the negative y-axis.
This statement is False. The proton experiences a force in the y-direction due to the magnetic field, but it also experiences a force in the x-direction due to the electric field. Therefore, the proton will accelerate in both the y and x directions, meaning its initial acceleration cannot be in the y-z plane alone.
C. When the proton enters the fields, the magnetic force on it is perpendicular to the electric field.
This statement is False. The magnetic force on the proton is given by the equation Fm = qvB, where v is the velocity of the proton and B is the magnetic field. Since v is in the positive x-direction and B is in the z-direction, the cross product of v and B will be in the negative y-direction. Therefore, the magnetic force is perpendicular to both the electric field and the proton's velocity.
D. The electric force on the proton is constant.
This statement is True. The electric force on a charged particle in a uniform electric field is constant, as long as the electric field remains constant.
E. The magnetic force on the proton is constant.
This statement is False. The magnetic force on a charged particle in a magnetic field depends on the velocity of the particle. Since the velocity of the proton is constant in this scenario, the magnetic force on the proton will also be constant.
F. The speed of the proton changes with time.
This statement is False. The speed of the proton, given as 10^4 m/s, remains constant throughout its motion. The forces acting on the proton only change its direction but do not affect its speed.
G. The path of the proton is a uniform helix.
This statement is True. When a charged particle moves through both an electric and magnetic field, it follows a curved path known as a helix. The path is helical because the magnetic force causes the particle to move in a circular path, while the electric force causes it to move along the z-axis. Thus, the path of the proton will indeed be a uniform helix.
To determine which of the statements are false, let's analyze each statement:
A. When the proton enters the fields, the magnitude of the magnetic force on it is ten times the magnitude of the electric force.
To calculate the magnitude of the electric force on the proton, we can use the formula F_e = qE, where F_e is the electric force, q is the charge of the proton, and E is the electric field intensity. Since the electric field is given as 10 V/m, we plug these values into the formula to find F_e = (1.6 x 10^-19 C)(10 V/m) = 1.6 x 10^-18 N.
The magnetic force on a charged particle moving at a velocity v within a magnetic field B can be calculated using the formula F_m = |q|vB, where F_m is the magnetic force, q is the charge of the particle, v is the velocity, and B is the magnetic field. Substituting the given values, we find F_m = (1.6 x 10^-19 C)(10^4 m/s)(10^-2 T) = 1.6 x 10^-17 N.
Therefore, the magnitude of the magnetic force is NOT ten times the magnitude of the electric force. Statement A is FALSE.
B. The initial acceleration of the proton is in the y-z plane, at 84 degrees to the z-axis and at 6 degrees to the negative y-axis.
To determine the initial acceleration of the proton, we need to consider the forces acting on it. In this case, we have both magnetic and electric forces.
Since the electric and magnetic fields are parallel to the z-axis, the only force that can cause acceleration in the y-z plane is the electric force. Therefore, the initial acceleration of the proton is solely in the z-axis direction, not in the y-z plane. Statement B is FALSE.
C. When the proton enters the fields, the magnetic force on it is perpendicular to the electric field.
Since the electric and magnetic fields are parallel to the z-axis, the direction of the magnetic force will also be in the z-axis direction. The electric field, however, is also parallel to the z-axis. Therefore, the magnetic force is NOT perpendicular to the electric field. Statement C is FALSE.
D. The electric force on the proton is constant.
From the given information, we can see that the electric field is uniform, meaning it has a constant magnitude and direction throughout the given region. Thus, the electric force acting on the proton is also constant. Statement D is TRUE.
E. The magnetic force on the proton is constant.
Similar to the electric force, the magnetic field is also uniform (has a constant magnitude and direction). Therefore, the magnetic force acting on the proton is constant. Statement E is TRUE.
F. The speed of the proton changes with time.
Since the forces acting on the proton are perpendicular to each other, the magnetic force does not affect the speed of the proton. Only the electric force can change its speed. As the electric force is constant and does not vary with time, the speed of the proton remains unchanged. Statement F is FALSE.
G. The path of the proton is a uniform helix.
Since the electric force does not affect the path of the proton (it only changes its speed), and the magnetic force acts perpendicular to the electric field, the path of the proton will be a straight line. Therefore, the path of the proton is NOT a uniform helix. Statement G is FALSE.
In conclusion, the false statements are A, B, C, F, and G.