Question

Two long thin parallel wires a distanced = 14.3 cm apart carry 25-A currents (1) in the same direction (out of the page). The wires have the same x-coordinate - one is directly above the other. The magnetic field is measured at point P, a distance dj = 13.4 cm from the lower wire and a distance d2 = 5.4 cm from the upper wire.

a. Find the x and y components of the magnetic field at P due to the current in the lower wire only.
b. Find the x and y components of the magnetic field at P due to the current in the upper wire only.
c. Determine the magnitude BI of the total magnetic field at P.

Answer 1

Answer: a. The x and y components of the magnetic field at point P due to the current in the lower wire only are B1x = (μ0 * I1) / (2π * d1) * cos(θ1) and B1y = (μ0 * I1) / (2π * d1) * sin(θ1).

b. The x and y components of the magnetic field at point P due to the current in the upper wire only can be calculated using the same formulas as in part a, but with the appropriate values for I2 and d2.

c. The magnitude of the total magnetic field at point P can be calculated using the formula BI = sqrt((B1x + B2x)^2 + (B1y + B2y)^2).

Explanation: Magnetic Field due to Current in Parallel Wires

When two long parallel wires carry currents, they create magnetic fields around them. The magnetic field at a point due to a current-carrying wire can be calculated using Ampere's Law.

Let's calculate the x and y components of the magnetic field at point P due to the current in the lower wire only (a).

Using Ampere's Law, the magnetic field at point P due to the current in the lower wire can be calculated as:

B1 = (μ0 * I1) / (2π * d1)

Where:

Using the given values, we can calculate the x and y components of B1:

B1x = B1 * cos(θ1)

B1y = B1 * sin(θ1)

Where:

Similarly, we can calculate the x and y components of the magnetic field at point P due to the current in the upper wire only (b).

Now, let's determine the magnitude of the total magnetic field at point P (c).

The total magnetic field at point P is the vector sum of the magnetic fields produced by each wire individually:

BI = sqrt((B1x + B2x)^2 + (B1y + B2y)^2)

Where:

Now, let's calculate the x and y components of the magnetic field at point P due to the current in the lower wire only (a).

Answer:

a. The x and y components of the magnetic field at point P due to the current in the lower wire only are:

B1x = (μ0 * I1) / (2π * d1) * cos(θ1)

B1y = (μ0 * I1) / (2π * d1) * sin(θ1)

b. The x and y components of the magnetic field at point P due to the current in the upper wire only can be calculated using the same formulas as in part a, but with the appropriate values for I2 and d2.

c. The magnitude of the total magnetic field at point P can be calculated using the formula:

BI = sqrt((B1x + B2x)^2 + (B1y + B2y)^2)