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.
T in the direction
To determine the magnitude and direction of the magnetic field (B), we can use the equations of motion for a charged particle moving in a magnetic field.
The force experienced by a charged particle (F) moving in a magnetic field (B) is given by the equation:
F = q * v * B * sin(θ)
Where:
- q is the charge of the particle (in this case, the charge of a proton is +e, where e is the elementary charge),
- v is the velocity of the particle,
- B is the magnitude of the magnetic field, and
- θ is the angle between the velocity vector and the magnetic field vector.
Since the proton experiences an acceleration in the +x direction when its velocity is in the +z direction, we can conclude that the angle between the velocity vector and the magnetic field vector is 90 degrees (perpendicular).
Now, we can rearrange the above equation to find the magnitude of the magnetic field (B):
B = F / (q * v * sin(θ))
Given:
- Acceleration (a) = 2.00 * 10^13 m/s^2
- Velocity (v) = 1.20 * 10^7 m/s
- Charge of a proton (q) = +e, where e is approximately 1.602 * 10^-19 C
Plugging in the values, we have:
B = (q * a) / (q * v * sin(θ))
Since sin(θ) = sin(90) = 1, we can simplify further:
B = a / v
Now, we can calculate the magnitude of the magnetic field:
B = 2.00 * 10^13 m/s^2 / 1.20 * 10^7 m/s
B = 1.67 * 10^6 T (Tesla)
The magnitude of the magnetic field is 1.67 * 10^6 Tesla.
To determine the direction of the magnetic field, we can use the right-hand rule. If the proton experiences an acceleration in the +x direction, then the magnetic field must be pointing in the -y direction. In other words, the magnetic field is perpendicular to both the velocity and acceleration vectors, pointing downward in the negative y-axis direction.
Therefore, the direction of the magnetic field is in the -y direction.