A proton moves perpendicular to a uniform magnetic field B at 1.20 107 m/s and experiences an acceleration of 2.00 1013 m/s2 in the +x direction when its velocity is in the +z direction. Determine the magnitude and direction of the field.
To determine the magnitude and direction of the magnetic field, we can use the equation for the magnetic force on a charged particle moving in a magnetic field.
The equation for the magnetic force is given by:
F = q * (v x B)
Where F is the magnetic force, q is the charge of the particle, v is the velocity of the particle, and B is the magnetic field.
In this case, we have a proton moving perpendicular to a uniform magnetic field. Since the proton has a positive charge, we can use the right-hand rule to determine the direction of the magnetic force. If the velocity of the proton is in the +z direction and the acceleration is in the +x direction, then the force on the proton must be in the +y direction (perpendicular to both the velocity and acceleration directions).
Now, let's solve for the magnitude of the magnetic field:
F = q * (v x B)
B = F / (q * (v x B))
First, we need to calculate the cross product (v x B):
v x B = | v | * | B | * sin(theta)
Since the velocity and magnetic field are perpendicular to each other, the angle theta is 90 degrees.
sin(theta) = sin(90) = 1
v x B = | v | * | B |
Next, we substitute this value into the equation for the magnitude of the field:
B = F / (q * (v x B))
B = F / (q * | v | * | B | )
Given that the acceleration is 2.00 * 10^13 m/s^2 and the velocity is 1.20 * 10^7 m/s, we can substitute these values into the equation:
B = (2.00 * 10^13) / (q * (1.20 * 10^7) )
Now, we need to find the charge of the proton. The charge of a proton is q = 1.60 * 10^(-19) C.
Substituting the values into the equation:
B = (2.00 * 10^13) / (1.60 * 10^(-19) * (1.20 * 10^7) )
Calculating the value, we find:
B ≈ 87.72 T
Therefore, the magnitude of the magnetic field is approximately 87.72 Tesla.
Since the force is in the +y direction, and the positive charge of the proton means it experiences a force in the direction of the magnetic field, we can conclude that the magnetic field is in the +y direction.