One long wire carries current 26.0 A to the left along the x axis. A second long wire carries current 68.0 A to the right along the line (y = 0.280 m, z = 0).
Where in the plane of the two wires is the total magnetic field equal to zero?
magnetic field is proportionalto I/r
so 0=26/y-68/(.280-y)
solve for y.
To find the point in the plane of the two wires where the total magnetic field is equal to zero, we can use the Biot-Savart law, which gives the magnetic field produced by a current-carrying wire at a certain point.
The magnetic field at a point P due to a current-carrying wire can be calculated using the formula:
B = (μ₀/4π) * (I * dl x r) / r²
Where:
B is the magnetic field
μ₀ is the permeability of free space (4π x 10^⁻7 T·m/A)
I is the current in the wire
dl is a small section of the wire through which the current flows
r is the displacement vector from the wire element dl to the point P
By using the Biot-Savart law, we can calculate the contributions to the magnetic field at any point due to both wires. The magnetic field at the point P in question will be the vector sum of the two contributions.
To simplify the problem, let's assume the wire carrying current 26.0 A is along the x-axis, and the wire carrying current 68.0 A is along the y-axis.
At any point (x, y, z) in the plane, the vector r = <x, y - 0.280, z - 0>.
We can divide the problem into two components: the x-component due to the wire along the x-axis and the y-component due to the wire along the y-axis.
For the x-component:
By considering a small section dl on the x-axis wire, dl = <dx, 0, 0>.
Substituting the values into the Biot-Savart law formula, we have:
dB_x = (μ₀/4π) * (26.0 * dx) / ((x-x₁)² + y² + (z-0.280)²)^(3/2)
For the y-component:
By considering a small section dl on the y-axis wire, dl = <0, dy, 0>.
Substituting the values into the Biot-Savart law formula, we have:
dB_y = (μ₀/4π) * (68.0 * dy) / (x² + (y - 0.280)² + z²)^(3/2)
To find the point where the total magnetic field is zero, we need to find a point (x, y, z) in the plane where the sum of the x-components and the sum of the y-components are both equal to zero, i.e., dB_x + dB_y = 0.
Solving this equation could be challenging, involving numerical methods or approximations, as it is a non-linear equation.