Agroup of particles is traveling in a magnetic field of unknown magnitude and direction. you observe that a proton moving at 1.70 km/s in the +x-direction experiences a force of 2.10×10^−16 n in the +y-direction, and an electron moving at 4.70 km/s in the −z-direction experiences a force of 8.40×10^−16 n in the +y-direction.

1-what is the magnitude of the magnetic field?
2-what is the direction of the magnetic field? (in the xz-plane)(degrees from the z direction)
3-what is the magnitude of the magnetic force on an electron moving in the −y-direction at 3.70 km/s ?
4-what is the direction of this the magnetic force? (in the xz-plane)(degrees from the x direction)

1)  B = 1,358 T , 2)  θ  = -55.35°, 3)   F = 8.04 10⁻¹⁶ N , 4) θ =  55.34°

Explanation:

The magnetic force is given by

         F = q v x B

Bold indicates vectors, one way to use this equation is to look for its magnitude

        F = q v B sin θ  

The direction is given by the right hand for a positive charge where the thumb points in the direction of the velocity, the fingers extended in the direction of the magnetic field and the palm in the direction of the force

Let's apply these relationships to this problem

.1) the magnitude of B

Let's use the proton data

         B = F / q v sin θ  

         B = 2.10 10⁻¹⁶/ (1.6 10⁻¹⁹ 1.70 10³  sin 90)

         B = 0.772 T

Let's perform the same calculation for the electron

         B = 8.40 10⁻¹⁶ / (1.6 10⁻¹⁹ 4.70 10³ sin 90)

         B = 1,117 T

We can see that there are two components of the magnetic field in the area

Let's look for the magnitude with the Pythagorean triangle

        B² = Bₓ² + ² + ²

        B = √(1,117² + 0.772²)

        B = 1,358 T

2) let's use the right hand rule.

For the proton

The thumb in the positive direction of the x axis (+ x)

The palm in the positive direction of the y axis (+ y)

Therefore the extended fingers point in the negative direction of z axis (-z) this is the direction of the magnetic field

For the electron

The thumb in the negative direction of the z axis (-z)

Since the electron has a negative charge, the direction of the force is contrary to the direction of the palm, in the negative direction of the y-axis (-y)

Therefore the extended fingers point in the positive direction of the x-axis (+ x)

Let's build the magnetic field vector

         B = Bₓ i ^ + j ^ + k ^

         B = 1,117 i ^ + 0 j ^ - 0.772 k ^

The field address can be found by trigonometry

         tan θ  = Bₓ /

         θ  = tan⁻¹ (Bₓ / )

         θ  = tan⁻¹ (1,117 / (- 0.772))

         θ  = -55.35°

3) the force for an electron

Another way to calculate the force is through determinants,

      F= -1.6 10⁻¹⁶

      F = -1.6 10-16 i (3.7 0.772) + k ^ (1,117 3.7)

      F = (-4.57 10-16 i ^ - 6.61 10-16 k ^) N

The magnitude is

    F = √ Fₓ² + ²

    F = √ (4.57² + 6.61²) 10⁻¹⁶

    F = 8.04 10⁻¹⁶ N

The address is

    tan θ = /

    θ = tan⁻¹ (6.61 / 4.57)

    θ =  55.34°