At the equator, the earth’s field is essentially horizontal; near the north pole, it is nearly vertical. In between, the angle varies. As you move farther north, the dip angle, the angle of the earth’s field below horizontal, steadily increases. Green turtles seem to use this dip angle to determine their latitude. Suppose you are a researcher wanting to test this idea. You have gathered green turtle hatchlings from a beach where the magnetic field strength is 5.0e-5 T and the dip angle is 56 degree. You then put the turtles in a 1.7 m diameter circular tank and monitor the direction in which they swim as you vary the magnetic field in the tank. You change the field by passing a current through a 250-turn horizontal coil wrapped around the tank. This creates a field that adds to that of the earth.
What current should you pass through the coil, to produce a net field in the center of the tank that has a dip angle of 62 degree?
To determine the current you need to pass through the coil to produce a net field in the center of the tank with a dip angle of 62 degrees, you can use the principles of magnetic fields and their orientations.
1. Start by understanding the dip angle and what it represents. The dip angle is the angle between the Earth's magnetic field and the horizontal plane at a specific location. It tells us the inclination or tilt of the Earth's magnetic field lines at that position.
2. The initial dip angle is given as 56 degrees, and you want to increase it to 62 degrees. This means you need to add a vertical component to the magnetic field in the center of the tank.
3. The dip angle is directly related to the vertical component of the magnetic field. The relationship is given by:
tangent(dip angle) = vertical component / horizontal component
Using the initial dip angle and the given magnetic field strength, you can calculate the horizontal component:
tangent(56 degrees) = vertical component / 5.0e-5 T
horizontal component = 5.0e-5 T / tangent(56 degrees)
4. Now, you need to find the magnetic field strength that will give you a dip angle of 62 degrees. Let's call this new magnetic field strength B_new.
tangent(62 degrees) = vertical component / horizontal component
vertical component = B_new * cos(62 degrees)
Substituting the expression for the horizontal component from step 3, you can solve for B_new:
tangent(62 degrees) = vertical component / (5.0e-5 T / tangent(56 degrees))
vertical component = B_new * cos(62 degrees)
B_new * cos(62 degrees) = tangent(62 degrees) * (5.0e-5 T / tangent(56 degrees))
B_new = tangent(62 degrees) * (5.0e-5 T / tangent(56 degrees)) / cos(62 degrees)
5. Finally, you can calculate the required current through the coil using Ampere's Law. Ampere's Law states that the magnetic field inside a solenoid is proportional to the current passing through it and the number of turns in the coil.
B_new = (μ₀ * N * I_new) / r
where B_new is the new magnetic field strength, μ₀ is the vacuum permeability, N is the number of turns in the coil, I_new is the new current you want to determine, and r is the radius of the coil.
Rearrange the equation to solve for I_new:
I_new = (B_new * r) / (μ₀ * N)
Substitute the values for B_new, r (radius of the tank), and N (number of turns in the coil) into the equation to find the required current.
Note: Make sure to use appropriate units while calculating.