Calculating Magnetic Flux: Plane with Rotated Wire and Current of 2.5 A

[hd28cw]

Find the magnetic flux crossing the portion of the plane
theta = x/4 defined by 0.01 m < r <0.05 m and 0 m < z < 2 m. A current of 2.50 A is flowing along z-axis along a very long wire.

in drawing the picture i know that there is a long thin wire with a current of 2.5 amps flowing positively on the z-axis and the plane is lying rotated at an angle of pi/4 with the magnetic field flowing in a counter clockwise direction.

How do I go about finding the magnetic flux.
Please help.

 

[Galileo]

Find the magnetic flux crossing the portion of the plane
theta = x/4 defined by 0.01 m < r <0.05 m and 0 m < z < 2 m.

Should that be theta=pi/4?


\Phi_B=\int_P \vec B \cdot d\vec a

Since the plane lies parallel with the z-axis, its normal points in the same direction as the magnetic field everywhere in the plane, so you can drop the dot in the integrand

\Phi_B=\int_P B da

Recall that the magnitude of the magnetic field of a long wire is:

B(r)=\frac{\mu_0 I}{2\pi r}
where r is the distance from the wire.

 

[wais]

To calculate the magnetic flux, we can use the formula Φ = BAcosθ, where Φ is the magnetic flux, B is the magnetic field, A is the area, and θ is the angle between the normal to the area and the magnetic field. In this case, we have a wire with a current of 2.5 A flowing along the z-axis, which creates a magnetic field around it. The plane is rotated at an angle of π/4, so the angle between the normal to the plane and the magnetic field is also π/4.

First, we need to find the magnetic field at the given point on the plane. Using the right-hand rule, we can determine that the direction of the magnetic field is out of the page. The magnitude of the magnetic field can be calculated using the formula B = μ0I/2πr, where μ0 is the permeability constant, I is the current, and r is the distance from the wire. In this case, r = 0.05 m. Plugging in the values, we get B = (4π×10^-7 T⋅m/A)(2.5 A)/(2π×0.05 m) = 0.001 T.

Next, we need to calculate the area of the portion of the plane defined by the given values of r and z. Since the plane is a circle, the area can be calculated using the formula A = πr^2. In this case, r = 0.05 m, so A = π(0.05 m)^2 = 0.00785 m^2.

Finally, we can plug in the values into the formula Φ = BAcosθ. Since θ = π/4, we get Φ = (0.001 T)(0.00785 m^2)cos(π/4) = 0.00000785 Wb. Therefore, the magnetic flux crossing the portion of the plane is 0.00000785 Wb.