An infinitely long wire carrying a current of 2 A is bent at a right angle as shown in the Figure. What is the magnetic field magnitude in Tesla a point P, 10 cm from the corner?
I can not draw pic here but I will try to describe it:
Point P is in right side straight from bend place. and y-axis (I mean from where it is bending ) is pointing down. X_axis is straight(as we know), and Y_axis is pointing down. Current I_1 in X_axis is flowing in opposite direction from point P, but I_2 in Y-axis is flowing upwards.
To solve this problem, we can use the Biot-Savart law, which states that the magnetic field created by a current-carrying wire at a point is directly proportional to the current and inversely proportional to the distance from the wire.
First, we need to find the magnetic field created by the wire segment carrying current I₁ in the x-axis.
Let's define a coordinate system where the y-axis is pointing downwards, the x-axis is horizontal and the z-axis is pointing out of the page. The wire segment carrying current I₁ is in the positive x-axis direction.
We can use the Biot-Savart law to find the magnetic field at point P due to this wire segment:
dB₁ = (μ₀/4π) * (I₁ * dl₁) / r₁²
Where:
- dB₁ is the magnetic field element created by the wire segment
- μ₀ is the permeability of free space (4π * 10^(-7) T·m/A)
- I₁ is the current in the wire segment
- dl₁ is a differential length element on the wire segment
- r₁ is the distance from the differential length element to point P
Since the wire segment is infinitely long, we only need to consider a small segment of it that is close to point P.
Now, let's find the magnetic field created by the wire segment carrying current I₂ in the y-axis.
dB₂ = (μ₀/4π) * (I₂ * dl₂) / r₂²
Where:
- dB₂ is the magnetic field element created by the wire segment
- I₂ is the current in the wire segment
- dl₂ is a differential length element on the wire segment
- r₂ is the distance from the differential length element to point P
Since the wires are perpendicular to each other, the net magnetic field at point P will be the sum of the magnetic fields created by the two wire segments.
B = √(B₁² + B₂²)
Now, to solve this problem, we need more information about the dimensions of the wire segments carrying current I₁ and I₂.
To determine the magnetic field magnitude at point P, we can use the right-hand rule for calculating the magnetic field due to a current-carrying wire.
Here's how you can calculate the magnetic field magnitude at point P:
1. Determine the magnetic field contribution due to the first segment of the wire (I1 in the X-axis):
- The magnetic field produced by a straight wire decreases with distance from the wire. The formula for the magnetic field magnitude at a distance r from a straight wire carrying current I is given by the equation:
B1 = (μ0 * I1)/(2π * r1), where μ0 is the permeability constant (4π * 10^-7 T∙m/A), I1 is the current in the first segment of the wire, and r1 is the distance of point P from the X-axis wire.
2. Determine the magnetic field contribution due to the second segment of the wire (I2 in the Y-axis):
- Similar to the first segment, use the same formula to calculate the magnetic field magnitude at point P caused by the second segment of the wire. However, this time consider the distance of point P from the Y-axis wire, which is 10 cm from the corner.
3. Combine the magnetic field contributions from both segments:
- Since the two magnetic field contributions are at right angles to each other, their vector sum can be found using the Pythagorean theorem. The total magnetic field magnitude at point P is given by the equation:
B = √(B1^2 + B2^2)
Plug in the values for the current in each segment (I1 and I2) as well as the distances of point P from each wire segment (r1 and r2). Apply the equations and calculate to find the magnetic field magnitude at point P.