Two straight conductors are 15 cm apart. If one conductor has a current of 12 A and the other has a current of 18 A, calculate the magnetic field at a point midway between the two conductors. The current in both conductors is flowing in the same direction.
To calculate the magnetic field at a point midway between the two conductors, we can use Ampere's Law. Ampere's Law states that the magnetic field around a closed loop is directly proportional to the current passing through the loop.
Here's how you can calculate the magnetic field:
1. Determine the magnetic field produced by each conductor individually.
- The magnetic field around a long straight conductor can be calculated using the formula:
B = (μ₀ * I) / (2π * r)
Where B is the magnetic field, μ₀ is the permeability of free space (4π × 10⁻⁷ Tm/A), I is the current in the conductor, and r is the distance from the conductor.
2. Calculate the magnetic field produced by each conductor at the point midway between the two conductors.
- For the first conductor with a current of 12 A, substitute the values into the formula:
B₁ = (μ₀ * I₁) / (2π * r₁)
- For the second conductor with a current of 18 A, substitute the values into the formula:
B₂ = (μ₀ * I₂) / (2π * r₂)
Since the two conductors are equidistant from the midpoint, r₁ = r₂ = 15 cm / 2 = 7.5 cm.
3. Add the magnetic fields produced by each conductor at the midpoint.
- B = B₁ + B₂
Let's compute the magnetic field at the midpoint using the given information:
For the first conductor with a current of 12 A:
B₁ = (4π × 10⁻⁷ Tm/A * 12 A) / (2π * 7.5 cm) = 4 × 10⁻⁷ T
For the second conductor with a current of 18 A:
B₂ = (4π × 10⁻⁷ Tm/A * 18 A) / (2π * 7.5 cm) = 6 × 10⁻⁷ T
Finally, adding the magnetic fields of both conductors:
B = B₁ + B₂ = 4 × 10⁻⁷ T + 6 × 10⁻⁷ T = 10 × 10⁻⁷ T = 10⁻⁶ T
Therefore, the magnetic field at the point midway between the two conductors is 10⁻⁶ Tesla, in the direction perpendicular to the plane formed by the two conductors.
To calculate the magnetic field at a point midway between the two conductors, you can use Ampere's law. Ampere's law states that the magnetic field around a closed loop is directly proportional to the current passing through the loop.
Here's how you can calculate the magnetic field at the midpoint:
Step 1: Determine the values given in the problem:
- Distance between the two conductors (d) = 15 cm = 0.15 m
- Current in the first conductor (I1) = 12 A
- Current in the second conductor (I2) = 18 A
Step 2: Calculate the magnetic field generated by each conductor at the midpoint separately.
The formula to calculate the magnetic field generated by a straight conductor at a point is given by:
B1 = (μ0 * I1) / (2π * r1)
Where:
B1 = Magnetic field generated by the first conductor
μ0 = Permeability of free space (constant value of 4π x 10⁻⁷ Tm/A)
I1 = Current in the first conductor
r1 = Distance from the first conductor to the midpoint
Similarly,
B2 = (μ0 * I2) / (2π * r2)
Where:
B2 = Magnetic field generated by the second conductor
I2 = Current in the second conductor
r2 = Distance from the second conductor to the midpoint
Step 3: Calculate the magnetic field at the midpoint by adding the magnetic fields generated by each conductor.
B_midpoint = B1 + B2
Since the conductors are 15 cm apart and the point of interest is halfway between them, the distances from each conductor to the midpoint are the same and will be half of the total distance i.e., r1 = r2 = d/2 = 0.15 m / 2 = 0.075 m.
Now, let's calculate the magnetic field:
B1 = (4π x 10⁻⁷ Tm/A * 12 A) / (2π * 0.075 m) = (4π x 10⁻⁷ * 12) / (2π * 0.075) = 5.108 x 10⁻⁶ T
B2 = (4π x 10⁻⁷ Tm/A * 18 A) / (2π * 0.075 m) = (4π x 10⁻⁷ * 18) / (2π * 0.075) = 7.662 x 10⁻⁶ T
B_midpoint = B1 + B2 = 5.108 x 10⁻⁶ T + 7.662 x 10⁻⁶ T = 1.177 x 10⁻⁵ T
Therefore, the magnetic field at the point midway between the two conductors is approximately 1.177 x 10⁻⁵ Tesla.