Question

Two infinite sheets of current flow parallel to the y-z plane as shown. The sheets are equally spaced from the origin by xo = 5.6 cm. Each sheet consists of an infinite array of wires with a density n = 19 wires/cm. Each wire in the left sheet carries a current I1 = 2.5 A in the negative z-direction. Each wire in the right sheet carries a current I2= 3.3 A in the positive z-direction. 1)What is Bx(P), the x-component of the magnetic field at point P, located at (x,y) = (-2.8 cm, 0)? ______T 2)What is By(P), the y-component of the magnetic field at point P, located at (x,y) = (-2.8 cm, 0)? _______T 3)What is By(R), the y-component of the magnetic field at point R, located at (x,y) = (-8.4 cm, 0)? _______T 4)What is ∮B⃗ ⋅dl⃗ where the integral is taken around the dotted path shown, from a to b to c to d to a. The path is a trapazoid with sides ab and cd having length 13.2 cm, side ad having length 6.1 cm, and side bc having length 9.7 cm. The height of the trapezoid is H = 13.1 cm. ______T-m 5)What is By(S), the y-component of the magnetic field at point S, located at (x,y) = ( 8.4 cm, 0)? _______T 6)What is ∫abB⃗ ⋅dl⃗ where the integral is taken along the dotted line shown, from a to b. _______T-m

Answer 1

Using the Biot-Savart law, we can calculate the magnetic field components at different points in the given setup. The x-component of the magnetic field at point P is zero due to the cancelation of contributions from the two wires. The y-components at points P, R, and S can be calculated using the Biot-Savart law. The line integrals around the trapazoid path and the path from a to b can be calculated by determining the magnetic field at each point and integrating the dot product along the path.

Step-by-step explanation:

To find the magnetic field components at point P, we can use the Biot-Savart law. Let's first calculate the x-component, Bx(P). Since the wire on the left carries current in the negative z-direction, it creates a magnetic field in the positive x-direction. The wire on the right, carrying current in the positive z-direction, creates a magnetic field in the negative x-direction. The contributions of these two wires cancel out, resulting in a net magnetic field of zero in the x-direction at point P.

For the y-component, By(P), both wires contribute to the magnetic field. The wire on the left creates a magnetic field in the positive y-direction, while the wire on the right creates a magnetic field in the negative y-direction. The net magnetic field in the y-direction at point P can be calculated using the Biot-Savart law.

By(R), the y-component of the magnetic field at point R, can be calculated using the same method. The distances from the wires are different, but the directions of the magnetic fields are the same.

∮B⃗ ⋅dl⃗ represents the line integral of the magnetic field around the given path. To calculate this, we need to determine the magnetic field at each point on the path and integrate the dot product of the magnetic field and the differential length element along the path. The magnetic field contributions from both wires need to be taken into account.

Similarly, By(S), the y-component of the magnetic field at point S, can be calculated using the Biot-Savart law.

∫abB⃗ ⋅dl⃗ represents the line integral of the magnetic field along the given path. To calculate this, we need to determine the magnetic field at each point on the path and integrate the dot product of the magnetic field and the differential length element along the path. The magnetic field contribution from the wire needs to be taken into account.

Answer 2

To find Bx(P), calculate the contributions from the left and right sheets by using the Biot-Savart law.

To find the x-component of the magnetic field at point P, we need to calculate the contributions from the left sheet (B1) and the right sheet (B2).

The magnetic field due to each sheet can be found using the Biot-Savart law. For the left sheet, B1 = (μ0 * n * I1)/2, where μ0 is the permeability of free space, n is the wire density, and I1 is the current in the left sheet. Using the given values, B1 = (4π × 10^-7 T·m/A * 19 wires/cm * 2.5 A)/2.

For the right sheet, B2 = (μ0 * n * I2)/(2 * xo), where xo is the spacing between the sheets. Using the given values, B2 = (4π × 10^-7 T·m/A * 19 wires/cm * 3.3 A)/(2 * 5.6 cm).

Ampère's law describes it and is expressed in terms of the line integral of B. The Biot-Savart law is another primary source of this law. We now examine that derivation for the unique scenario of an infinite, linear wire. →B⋅d→l=Brdθ.

Both aid in the spreading of electric fields. Ampere's law, on the other hand, works best with symmetric fields, whereas Biot-Savart's method makes it possible to determine the field for any given object, however it is a far more difficult computation.