A proton moves with a velocity of vector v = (6i hat bold - 3j hat bold + k hat bold) m/s in a region in which the magnetic field is vector B = (i hat bold + 2j hat bold - 3k hat bold) T. What is the magnitude of the magnetic force this charge experiences? N
v=(6i-3j+k) m/s
B=(i+2j-3k) T
F=e[v,B]
i j k
x₁ y₁ z₁ =
x₂ y₂ z₂
i j k
6 -3 1 =
1 2 -3
=(y₁z₂-z₁y₂)i +(z₁x₂-x₁z₂)j +(x₁y₂-y₁x₂)k=
=(7i+19j+15k).
|[v,B] |=sqrt{7²+19²+15²}=25.2
F= 1.6•10⁻¹⁹•25.2=4.03•10⁻¹⁸ J.
To find the magnitude of the magnetic force experienced by a charged particle moving in a magnetic field, you can use the formula:
F = q * v * B * sin(theta)
Where:
- F represents the magnitude of the magnetic force
- q is the charge of the particle
- v is the velocity vector of the particle
- B is the magnetic field vector
- theta is the angle between the velocity vector and the magnetic field vector
In this case, we have:
q = charge of a proton = 1.6 * 10^-19 C (Coulombs)
v = velocity vector = 6i - 3j + k (m/s)
B = magnetic field vector = i + 2j - 3k (T)
theta = angle between v and B
To find the angle theta, you can take the dot product of the velocity vector and the magnetic field vector:
v · B = |v| |B| cos(theta)
v · B = (6i - 3j + k) · (i + 2j - 3k)
= 6*1 + (-3)*2 + (1)*(-3)
= 6 - 6 - 3
= -3
|v| = √(6^2 + (-3)^2 + 1^2)
= √(36 + 9 + 1)
= √46
|B| = √(1^2 + 2^2 + (-3)^2)
= √(1 + 4 + 9)
= √14
Therefore, the magnitude of the angle theta is given by:
cos(theta) = (-3) / (√46 * √14)
theta = arccos((-3) / (√46 * √14))
Now, we can calculate the magnitude of the magnetic force:
F = q * v * B * sin(theta)
= (1.6 * 10^-19 C) * (6i - 3j + k) * (i + 2j - 3k) * sin(theta)
To simplify the expression, you can calculate the cross product of v and B:
v x B = |i j k |
|6 -3 1 |
|1 2 -3|
= (6*(-3) - (-3)*2)i + (6*1 - 1*(-3))j + ((-3)*2 - 1*6)k
= (-18 + 6)i + (6 + 3)j + (-6 - 6)k
= -12i + 9j - 12k
Finally, we can calculate the magnitude of the magnetic force:
F = q * |v| * |B| * sin(theta)
= (1.6 * 10^-19 C) * √46 * √14 * sin(theta)
Now, substitute the value of sin(theta) and calculate the final answer.