A proton is traveling horizontally at 5.4 × 106 m/s in a direction 45o north of west. A magnetic flux density has a magnitude of 0.25 T and is directed north. What is the force on the proton?
Answer 1.5 × 10-13 N, up
2.1 × 10-13 N, down
1.5 × 10-13 N, down
2.1 × 10-13 N, up
1.5
To calculate the force on a charged particle moving in a magnetic field, we use the equation:
F = q * (v x B)
Where:
- F is the force on the particle,
- q is the charge of the particle,
- v is the velocity vector of the particle,
- B is the magnetic flux density vector.
In this case, we are given:
- q = charge of a proton = +1.6 x 10^-19 C (Coulombs)
- v = velocity of the proton = 5.4 x 10^6 m/s at a direction 45° north of west
- B = magnetic flux density = 0.25 T (Tesla) directed north.
First, we need to determine the velocity vector, v.
The velocity v is given as 5.4 x 10^6 m/s at an angle of 45° north of west. Let's break it down into its horizontal and vertical components.
The horizontal component, vx, can be determined using cos(45°) = vx / (5.4 x 10^6 m/s):
vx = (5.4 x 10^6 m/s) * cos(45°)
The vertical component, vy, can be determined using sin(45°) = vy / (5.4 x 10^6 m/s):
vy = (5.4 x 10^6 m/s) * sin(45°)
Now, we have the velocity vector in terms of its horizontal and vertical components:
v = vx i + vy j
Next, we can calculate the cross product (v x B) to find the direction of the force.
The cross product between two vectors, A and B, can be determined using the following formula:
A x B = (Ay * Bz - Az * By) i + (Az * Bx - Ax * Bz) j + (Ax * By - Ay * Bx) k
In this case, A = v and B = B.
Using the cross product formula, we can calculate (v x B).
Finally, we can calculate the force, F, using the formula F = q * (v x B).
Plug in the values of q, v, and (v x B) into the formula and calculate F.
The force on the proton is the resulting force vector and can be expressed as a combination of its horizontal and vertical components. The direction can be determined by analyzing the signs of each component.