At a certain location on the Earth,

the horizontal component of the Earth’s magnetic field is 2.5 ·10−5 T due North. A proton moves
eastward with just the right speed for the magnetic force on it to balance its weight so the net
force on the proton will be zero. Find the speed of the proton for this to occur.

e V B = m g

since the directions of V and B are perpendicular

m is the proton mass and e is its charge. g is the acceleration of gravity

Solve for V

To find the speed of the proton in order for the magnetic force on it to balance its weight, we need to understand the relationship between the magnetic force and the motion of a charged particle in a magnetic field.

The magnetic force on a charged particle moving in a magnetic field can be calculated using the equation:

F = q * v * B * sin(theta)

Where:
- F is the magnetic force
- q is the charge of the particle
- v is the velocity of the particle
- B is the magnetic field strength
- theta is the angle between the velocity and the magnetic field

In this case, since the net force on the proton should be zero, we can set the magnetic force equal to the weight of the proton.

mg = q * v * B * sin(theta)

Since we know the magnitude of the horizontal component of the Earth's magnetic field is 2.5 × 10^(-5) T due North, we can assume that the angle between the velocity and the magnetic field (theta) is 90 degrees, because the particle is moving eastward (perpendicular to the magnetic field).

Now, we can rearrange the equation to solve for the velocity (v):

v = (mg) / (q * B * sin(theta))

The mass of a proton (m) is approximately 1.67 × 10^(-27) kg, and the charge of a proton (q) is 1.6 × 10^(-19) C.

Plugging in these values, we have:

v = ((1.67 × 10^(-27) kg) * (9.8 m/s^2)) / ((1.6 × 10^(-19) C) * (2.5 × 10^(-5) T) * 1)

Simplifying the equation, we get:

v = (1.65152 × 10^(-26) kg m/s) / (4 × 10^(-24) N/C)

Finally, dividing these values, we can find the speed of the proton:

v ≈ 4.128 × 10^(-3) m/s

Therefore, the speed of the proton for the magnetic force to balance its weight is approximately 0.004128 m/s.