1.The earth is constantly bombarded by cosmic rays, which are charged particles from outer space. moving vertically Find the force on one such cosmic ray− a proton that is instantaneously downward at 5.04 107 m/s at a point outside the atmosphere over New York, where the magnetic field dips 74.0° below the horizontal and has a horizontal component of 1.70 10-5 T
2.Stainless-steel poles were arranged by an artist for the purpose of producing an enhanced display of lightning in a field of nearly one square mile. The magnetic field in New Mexico, where the rods are located, dips 64.0° below the horizontal and has a horizontal component of magnitude 2.5 10-5 T. Calculate the magnetic force on one of these poles, 6.63 m long, when it transmits an upward current of 1.98 104 A during a lightning discharge.
F = q V x B where the "x" indicates a vector product.
We will be glad to critique your work. \.
13.2*10^-17
To find the force on a charged particle, such as a proton or a pole, in a magnetic field, you can use the formula for the magnetic force:
F = q * v * B * sin(θ)
Where:
F is the magnitude of the magnetic force
q is the charge of the particle
v is the velocity of the particle
B is the magnetic field strength
θ is the angle between the velocity vector and the magnetic field vector
Let's calculate the force for each scenario step-by-step:
1. Cosmic Ray:
Given:
Velocity of the proton, v = 5.04 * 10^7 m/s
Magnetic field horizontal component, B = 1.70 * 10^-5 T
Angle, θ = 74° below the horizontal
In this case, we have a downward velocity, so the angle between the velocity and the magnetic field is 180° - θ.
θ = 74°
Angle between velocity and magnetic field = 180° - 74° = 106°
Now, let's calculate the force using the formula:
F = q * v * B * sin(θ)
Since the proton has a charge equal to the elementary charge, q = 1.602 * 10^-19 C, we can substitute the values into the formula:
F = (1.602 * 10^-19 C) * (5.04 * 10^7 m/s) * (1.70 * 10^-5 T) * sin(106°)
Calculating:
F = 1.602 * 5.04 * 1.70 * sin(106°) * 10^-19 * 10^7 * 10^-5
F = 13.6133 * 10^-17 N
Therefore, the force on the proton is approximately 1.36 * 10^-16 N.
2. Lightning Pole:
Given:
Length of the pole, l = 6.63 m
Current, I = 1.98 * 10^4 A
Magnetic field horizontal component, B = 2.5 * 10^-5 T
Angle, θ = 64° below the horizontal
Let's calculate the force using the formula:
F = q * v * B * sin(θ)
In this case, the velocity of the charged particles (electrons) in the pole is the drift velocity, which can be calculated as:
v = I / (n * A * e)
Where:
n is the number density of charged particles
A is the cross-sectional area of the pole
e is the elementary charge
The drift velocity depends on the material and the dimensions of the pole. Let's assume a typical value for the drift velocity in stainless steel, approximately v = 1 * 10^-3 m/s.
Now, let's calculate the force using the formula:
F = q * v * B * sin(θ)
Since the pole is neutral and made of stainless steel, q (charge) is zero, meaning the force will also be zero.
Therefore, for the lightning pole, the magnetic force is zero.
To find the force on a charged particle moving in a magnetic field, you can use the equation:
F = q * v * B * sin(θ)
Where:
F = force exerted on the particle
q = charge of the particle
v = velocity of the particle
B = magnetic field
θ = angle between the velocity vector and the magnetic field vector
Let's calculate the force for each question:
Question 1:
The proton is moving downward with a velocity of 5.04 × 10^7 m/s. The magnetic field has a horizontal component of 1.70 × 10^-5 T, and the angle of the field with the horizontal is 74.0° below.
To find the force on the proton, we need to consider the charge of the proton. The charge on a proton is +1.6 × 10^-19 C. Substituting the given values into the equation, we have:
F = (1.6 × 10^-19 C) * (5.04 × 10^7 m/s) * (1.70 × 10^-5 T) * sin(74.0°)
Calculate the sin(74.0°) using a calculator. Then plug in the values and calculate the force.
Question 2:
The stainless-steel pole is transmitting an upward current of 1.98 × 10^4 A. The magnetic field has a horizontal component of magnitude 2.5 × 10^-5 T, and the angle of the field with the horizontal is 64.0° below.
To find the force on the pole, we need to consider the length of the pole and the current passing through it. The length of the pole is 6.63 m, and the charge carriers in the pole are electrons. The charge of an electron is -1.6 × 10^-19 C. The total charge passing through the pole can be calculated by multiplying the current with time.
Substituting the given values into the equation, we have:
F = (-1.6 × 10^-19 C) * (1.98 × 10^4 A) * (6.63 m) * (2.5 × 10^-5 T) * sin(64.0°)
Calculate the sin(64.0°) using a calculator. Then plug in the values and calculate the force.
Remember to double-check the values and units before plugging them into the equation to ensure accuracy.