A long thin copper wire has a mass of 100 grams / meter. If it is arranged near the surface of the earth pointing north in a 0.5 Tesla magnetic field pointing west, what current (in amps) is needed to levitate the rod (against gravity)?
Consider the gravitational and magnetic forces on a 1 meter section of wire. The magnetic force is I∗(1m)∗(0.5T) since the current and field are perpendicular, and the gravitational force is (0.1kg)∗(9.81m/s2). To levitate, these forces must be equal. Equate the expressions and solve for current.
To determine the current needed to levitate the copper wire, we can use the formula for the force on a current-carrying wire in a magnetic field.
The formula for the force on a current-carrying wire in a magnetic field is given by:
F = BILsinθ
Where:
F is the force on the wire,
B is the magnetic field strength,
I is the current flowing through the wire,
L is the length of the wire, and
θ is the angle between the magnetic field and the wire.
In this case, we want to find the current (I) required to levitate the wire against gravity. So, the force we are interested in is the force of gravity acting on the wire.
The force of gravity can be calculated using the equation:
F_gravity = mg
Where:
F_gravity is the force of gravity,
m is the mass of the wire per unit length (100 grams/meter), and
g is the acceleration due to gravity near the surface of the Earth (approximately 9.8 m/s^2).
Now, since the wire is arranged near the surface of the earth pointing north and the magnetic field is pointing west, we can determine that the angle (θ) between the magnetic field and the wire is 90 degrees.
So, we can set up the equation equating the magnetic force to the force of gravity:
BILsinθ = mg
Substituting the given values:
(0.5 Tesla)(I)(1 meter)(sin 90°) = (0.1 kg)(9.8 m/s^2)
Simplifying the equation:
0.5 I = 0.1 * 9.8
0.5 I = 0.98
Dividing both sides of the equation by 0.5:
I = 0.98 / 0.5
I ≈ 1.96 Amps
Therefore, approximately 1.96 amps of current is needed to levitate the copper wire against gravity in the given magnetic field.