Two long parallel wires carry currents of 10 A in opposite directions. they are separated by 40 cm. what is the magnetic field in the plane of the wires that is 20 cm from one wire and 60 cm from the other?

An object 7.25 cm high is placed 15.1 cm from a convex mirror of radius 55.2cm. (a) What is the image distance?

To find the magnetic field at a point between two long parallel wires carrying currents, you can use the Biot-Savart law. The formula for the magnetic field due to a long wire is given by:

B = (μ₀ * I) / (2π * r)

Where:
B is the magnetic field
μ₀ is the permeability of free space (4π * 10^-7 T*m/A)
I is the current in the wire
r is the distance from the wire

In this case, we have two wires carrying currents in opposite directions. The magnetic fields due to each wire will have opposite directions and will add up at the point of interest.

Let's calculate the magnetic field due to each wire separately and then take their sum at the end.

First, let's calculate the magnetic field due to the wire that is 20 cm away from the point of interest:

B₁ = (μ₀ * I₁) / (2π * r₁)

Where:
I₁ = 10 A (current in the wire)
r₁ = 20 cm = 0.2 m (distance from the wire)

Plugging in the values:

B₁ = (4π * 10^-7 T*m/A * 10 A) / (2π * 0.2 m)
B₁ = 10^-6 T

Next, let's calculate the magnetic field due to the wire that is 60 cm away from the point of interest:

B₂ = (μ₀ * I₂) / (2π * r₂)

Where:
I₂ = -10 A (opposite direction since the currents are in opposite directions)
r₂ = 60 cm = 0.6 m

Plugging in the values:

B₂ = (4π * 10^-7 T*m/A * -10 A) / (2π * 0.6 m)
B₂ = -10^-6 T

Now, we can find the total magnetic field at the point of interest by summing the fields due to each wire:

B_total = B₁ + B₂
B_total = 10^-6 T + (-10^-6 T)
B_total = 0 T

Therefore, the magnetic field at the point between the two wires is zero.

To find the magnetic field at a point in the plane of the wires, you can use Ampere's Law. Ampere's Law states that the magnetic field around a closed loop is directly proportional to the total electric current passing through the loop.

To apply Ampere's Law to this problem, follow these steps:

Step 1: Determine the loop you want to consider. In this case, you can consider a rectangular loop that encloses the point of interest, with two sides parallel to the wires and two sides connecting them.

Step 2: Calculate the total current passing through the loop. In this case, since the two wires carry currents of 10 A in opposite directions, the total current passing through the loop is the difference between the two currents, which is 10 A - (-10 A) = 20 A.

Step 3: Calculate the length of the loop. The length of the loop is the sum of the distances from the point of interest to each wire, which is 20 cm + 60 cm = 80 cm = 0.8 m.

Step 4: Apply Ampere's Law. According to Ampere's Law, the magnetic field (B) multiplied by the length of the loop (L) is equal to the permeability of free space (μ₀) multiplied by the total current passing through the loop (I). In equation form: B * L = μ₀ * I.

Rearrange the equation to solve for the magnetic field B: B = (μ₀ * I) / L.

Step 5: Substitute the known values into the equation. The permeability of free space (μ₀) is approximately 4π × 10^(-7) T·m/A.

B = (4π × 10^(-7) T·m/A * 20 A) / 0.8 m

Step 6: Calculate the magnetic field B using the values from Step 5.

B = 4π × 10^(-7) T·m/A * 25 A = 10π × 10^(-6) T = 3.14 × 10^(-5) T.

Therefore, the magnetic field in the plane of the wires that is 20 cm from one wire and 60 cm from the other is approximately 3.14 × 10^(-5) T.