a long horizontal wire is rigidly placed, carrying a current of 50a. another parallel long wire carrying a current of 40a is just supported above it.determine the distance between them at which the upper wire can be just supported by magnetic repulsion itself.(weight of upper wire=0.05 N/m)
To determine the distance between the two wires at which the upper wire can be just supported by magnetic repulsion, we can use the magnetic force formula:
F = μ₀ * I₁ * I₂ * L / (2πd)
Where:
F is the magnetic force between the wires,
μ₀ is the permeability of free space (4π * 10^(-7) T * m / A),
I₁ and I₂ are the currents in the wires (50 A and 40 A, respectively),
L is the length of the wires in meters (we assume this is sufficiently long),
and d is the distance between the wires.
In this case, we want to find the distance d, when the magnetic force F is equal to the weight per unit length (0.05 N/m) of the upper wire.
Setting up the equation:
0.05 N/m = (4π * 10^(-7) T * m / A) * (50 A) * (40 A) * L / (2πd)
Cancelling out common terms:
0.05 = (4π * 10^(-7) * 50 * 40 * L) / (2 * d)
Simplifying further:
0.05 = (4 * 3.14 * 10^(-7) * 50 * 40 * L) / d
Cross-multiplying and rearranging:
d = (4 * 3.14 * 10^(-7) * 50 * 40 * L) / 0.05
Calculating the value of d:
d ≈ 16 m
Therefore, the distance between the wires at which the upper wire can be just supported by magnetic repulsion itself is approximately 16 meters.
To determine the distance between the wires at which the upper wire can be just supported by magnetic repulsion, we can use the formula for the magnetic force between two current-carrying wires.
The formula for the magnetic force between two parallel wires is given by:
F = (μ₀ * I₁ * I₂ * L) / (2π * d)
Where:
F = Magnetic force
μ₀ = Magnetic constant (4π * 10^-7 T·m/A)
I₁ = Current in the first wire
I₂ = Current in the second wire
L = Length of the wires
d = Distance between the wires
In this case, we have:
I₁ = 50 A (current in the lower wire)
I₂ = 40 A (current in the upper wire)
L = length of the wires (not provided)
We need to determine the distance, d, at which the magnetic force is equal to the weight of the upper wire. Let's assume we are given the length of the wires, L.
Step 1: Calculate the magnetic force
Since we know the current and length of the wires, we can calculate the magnetic force using the formula above:
F = (4π * 10^-7 T·m/A) * (50 A) * (40 A) * L / (2π * d)
F = (8π² * 10^-7) * (2000) * L / (d)
Step 2: Set up the equation
Now we equate the magnetic force to the weight of the upper wire to find the distance at which the wires are just supported:
F = Weight of the upper wire
(8π² * 10^-7) * (2000) * L / (d) = 0.05 N/m
Step 3: Solve for the distance (d)
Rearrange the equation to solve for d:
d = (8π² * 10^-7) * (2000) * L / (0.05)
Evaluate the expression on the right-hand side using known values for π, μ₀, and L to find the distance, d.