Final answer:
The magnitude of the horizontal component of the Earth's magnetic field is approximately 4.03 x 10^-5 T. Using the right-hand rule, the east end of the aluminum bar is determined to be the positive side as the bar falls through the Earth's magnetic field.
Step-by-step explanation:
To determine the magnitude of the horizontal component of the Earth's magnetic field, we use Faraday's law of electromagnetic induction. As the aluminum bar moves through the Earth's magnetic field, an electromotive force (emf) is induced across it.
The induced emf (ε) in a conductor moving through a magnetic field is given by ε = Bvl sin(θ), where B is the magnetic field strength, v is the velocity of the conductor, l is the length of the conductor, and θ is the angle between the velocity and the magnetic field. Since the aluminum bar is moving vertically through the Earth's magnetic field which is horizontal and the bar is parallel to the east-west direction, θ is 90 degrees making sin(θ) = 1.
Therefore, the formula simplifies to ε = Bvl. Rearranging this formula to solve for B gives us B = ε / (vl). Substituting the given values, we have B = (7.6 × 10^-4 V) / (28 m/s × 0.67 m). Calculating this gives us B as approximately 4.03 × 10^-5 T.
To determine which end of the bar is positive, we apply the right-hand rule. If we point our thumb in the direction of the velocity (downward), and our fingers in the direction of the magnetic field (north), the palm faces towards the side that becomes positive, which is the east side in this case. Hence, the east end of the bar is positive.
The Complete Queston is:
A 0.67-m aluminum bar is held with its length parallel to the east-west direction and dropped from a bridge. Just before the bar hits the river below, its speed is 28 m/s, and the emf induced across its length is 7.6 x 10-4 V. Assuming the horizontal component of the Earth's magnetic field at the location of the bar points directly north, (a) determine the magnitude of the horizontal component of the Earth's magnetic field, and (b) state whether the east end or the west end of the bar is positive.?