Near Earth, the density of protons in the solar wind (a stream of particles from the Sun) is 12.4 cm-3, and their speed is 576 km/s. (a) Find the current density of these protons. (b) If Earth's magnetic field did not deflect the protons, what total current in amperes Earth receive?
change km/s to cm/s, then multiply density given times speed.
Current = current density*area crosssection earth/charge per proton
For part (b), multiply the current density (from part a) by pi R^2, where R is the radius of the Earth.
To find the current density of the protons, we need to calculate the number of protons passing through a unit area per unit time. The formula for current density is given by:
Current Density = Charge Density x Velocity
(a) So, first we need to find the charge density of the protons. The charge of a proton (e) is 1.6 x 10^-19 C. The mass of a proton is approximately 1.67 x 10^-27 kg.
The charge density is the product of the proton density and the charge of one proton:
Charge Density = Proton Density x Charge of a Proton
Charge Density = 12.4 cm^-3 x (1.6 x 10^-19 C)
Now, let's convert the charge density to units of C/m^3:
Charge Density = 12.4 x (1.6 x 10^-19 C) / (1 cm^3) * (1 m / 100 cm)^3
Next, we can compute the current density:
Current Density = Charge Density x Velocity
Current Density = Charge Density x (576,000 m/s)
(b) To determine the total current Earth receives, we multiply the current density by the area over which the protons pass:
Total Current = Current Density x Area
The area over which the protons pass is the cross-sectional area of Earth, which is equal to πr^2, where r is the radius of Earth. The radius of Earth is approximately 6,371 km, so let's convert it to meters:
Radius = 6,371 km x 1000 m / 1 km = 6,371,000 m
Now, we can calculate the total current Earth receives:
Total Current = Current Density x (π x (6,371,000 m)^2)
Finally, to obtain the answer in units of amperes (A), we divide by the elementary charge (e):
Total Current (in Amperes) = Total Current / Elementary Charge
Now, you can substitute the values into the equations and perform the necessary calculations.