Suppose a straight 1.60mm -diameter copper wire could just "float" horizontally in air because of the force due to the Earth's magnetic field B⃗ , which is horizontal, perpendicular to the wire, and of magnitude 4.2×10−5T .
What current would the wire carry?
To determine the current that the wire would carry, we can use the formula for the magnetic force experienced by a current-carrying wire:
F = BIL
Where:
F is the magnetic force,
B is the magnetic field strength,
I is the current flowing through the wire, and
L is the length of the wire.
In this scenario, the wire is floating horizontally, which means the magnetic force experienced by the wire is balanced by the gravitational force acting on it. Therefore, we can equate these two forces:
F_magnetic = F_gravitational
The magnetic force can be calculated using the formula mentioned earlier, and the gravitational force acting on the wire can be calculated using the formula:
F_gravitational = mg
Where:
m is the mass of the wire, and
g is the acceleration due to gravity.
Since the wire is in equilibrium, we can set the magnetic force equal to the gravitational force:
BIL = mg
Rearranging the equation, we get:
I = mg / (BL)
We have the following values:
Wire diameter (d) = 1.60 mm
Wire radius (r) = d/2
Wire length (L) = Unknown
Magnetic field strength (B) = 4.2×10^−5 T
Acceleration due to gravity (g) = 9.8 m/s^2
Density of copper (ρ) = 8,960 kg/m^3
To determine the length of the wire, we need to make an assumption about its mass per unit length (linear mass density). For copper wire, it is common to use a value of 8.96 g/cm^3.
Using the equation for linear mass density:
ρ = m / L
Where:
ρ is the linear mass density,
m is the mass of the wire, and
L is the length of the wire.
Rearranging the equation, we get:
m = ρL
Substituting the value of ρ = 8.96 g/cm^3 and converting it to kg/m^3, we have:
ρ = 8.96 × 10^3 kg/m^3
By multiplying the area of the wire by its density, we can calculate the mass per unit length:
m = ρπr^2
Finally, the equation for the current becomes:
I = m * g / (B * L)
Substituting the known values into the equation will provide the current flowing through the wire.