One component of a magnetic field has a magnitude of 0.0445 T and points along the +x axis, while the other component has a magnitude of 0.0725 T and points along the -y axis. A particle carrying a charge of +1.01 × 10-5 C is moving along the +z axis at a speed of 4.18 × 103 m/s. (a) Find the magnitude of the net magnetic force that acts on the particle. (b) Determine the angle that the net force makes with respect to the +x axis.
B=sqrt{B(x)²+B(y)²} = …
F=qvBsinα
sinα=sin90=1 =>
F=qvB = …
To find the magnitude of the net magnetic force that acts on the particle, we can use the formula:
F = q(v x B)
where F is the force, q is the charge of the particle, v is the velocity of the particle, and B is the magnetic field.
a) First, let's find the vector product of v and B:
v x B = |v| |B| sin(theta) n
where |v| is the magnitude of velocity, |B| is the magnitude of the magnetic field, theta is the angle between v and B, and n is the unit vector perpendicular to the plane formed by v and B.
Since the particle is moving along the +z axis, the angle theta is 90 degrees, and sin(theta) equals 1. Therefore,
v x B = |v| |B| n
Now, let's calculate the magnitude of the net magnetic force:
F = q(v x B)
Given:
q = +1.01 × 10^-5 C (charge of the particle)
|v| = 4.18 × 10^3 m/s (magnitude of velocity)
|B| = 0.0445 T (magnitude of the magnetic field)
F = (+1.01 × 10^-5 C) (4.18 × 10^3 m/s) (0.0445 T)
Calculate the product: (+1.01 × 10^-5 C) (4.18 × 10^3 m/s) = 4.2178 × 10^-2 N
Now, multiply this by 0.0445 T:
F = (4.2178 × 10^-2 N) (0.0445 T)
Calculate the product: (4.2178 × 10^-2 N) (0.0445 T) = 1.880421 × 10^-3 N
Therefore, the magnitude of the net magnetic force that acts on the particle is approximately 1.880421 × 10^-3 N.
b) To determine the angle that the net force makes with respect to the +x axis, we can use trigonometry.
Let's call this angle phi.
tan(phi) = (By/Bx)
where By is the y-component of the magnetic field and Bx is the x-component of the magnetic field.
Given:
By = 0.0725 T
Bx = 0.0445 T
tan(phi) = (0.0725 T)/(0.0445 T)
Now, calculate the ratio: (0.0725 T)/(0.0445 T) ≈ 1.629
Therefore, the angle phi is approximately tan^(-1)(1.629), which is approximately 58.798 degrees.
Hence, the angle that the net force makes with respect to the +x axis is approximately 58.798 degrees.