A long wire carrying a 5.0 A current perpendicular to the xy-plane intersects the x-axis at x = -2.0 cm. A second, parallel wire carrying a 2.5 A current intersects the x-axis at x = 2.0 cm.

PART A
At what point on the x-axis is the magnetic field zero if the two currents are in the same direction?

PART B
At what point on the x-axis is the magnetic field zero if the two currents are in opposite directions?

I don't understand !!! after Physics.

part A is biot-savart problem, add the two magnetic fields to be zero, you know the total distance, currents.

partB. Same law. You know it wont be between the wires.

PART A:

To find the point on the x-axis where the magnetic field is zero when the two currents are in the same direction, we can use the formula for the magnetic field due to a long straight wire. This formula is given by:

B = (μ₀ * I) / (2πr),

where B is the magnetic field, μ₀ is the permeability of free space (equal to 4π × 10^-7 T·m/A), I is the current, and r is the distance from the wire.

Considering the first wire carrying a current of 5.0 A, and using x = -2.0 cm = -0.02 m as the intersection point on the x-axis, we can find the magnetic field produced by this wire at this point. Plugging the values into the formula, we have:

B₁ = (4π × 10^-7 T·m/A * 5.0 A) / (2π *(-0.02 m)).

Simplifying this expression, we get:

B₁ = -(μ₀ * I) / (2r).

Now, considering the second wire carrying a current of 2.5 A, and using x = 2.0 cm = 0.02 m as the intersection point on the x-axis, the magnetic field produced by this wire at this point can be found similarly:

B₂ = (4π × 10^-7 T·m/A * 2.5 A) / (2π * 0.02 m).

Simplifying this expression, we get:

B₂ = (μ₀ * I) / (2r).

Since the two currents are in the same direction, the magnetic fields produced by the two wires will have opposite directions. So, to find the point on the x-axis where the magnetic field is zero, we need to equate the two magnetic fields and solve for r:

B₁ = -B₂.

-(μ₀ * I) / (2r) = (μ₀ * I) / (2r).

Simplifying this expression and cross-multiplying, we have:

-(μ₀ * I) = μ₀ * I.

Now, canceling out μ₀ and dividing both sides by I, we get:

-1 = 1.

However, this equation is not possible, which means there is no point on the x-axis where the magnetic field is zero when the two currents are in the same direction.

PART B:
To find the point on the x-axis where the magnetic field is zero when the two currents are in opposite directions, we can use the same approach as in Part A.

The magnetic fields produced by the two wires will have the same direction since the currents are in opposite directions. So, we need to equate the two magnetic fields again and solve for r:

B₁ = B₂.

-(μ₀ * I) / (2r) = (μ₀ * I) / (2r).

Cross-multiplying and simplifying, we have:

-(μ₀ * I) = μ₀ * I.

Canceling out μ₀ and dividing both sides by I, we get:

-1 = 1.

Again, this equation is not possible, which means there is no point on the x-axis where the magnetic field is zero when the two currents are in opposite directions.

To find the point on the x-axis where the magnetic field is zero, we can use the Biot-Savart Law, which relates the magnetic field around a current carrying wire to the magnitude and direction of the current.

PART A:
If the currents in the two wires are in the same direction, we know that they both create a magnetic field that circles around the wires in the same direction. To find the point where their magnetic fields cancel out, we need to calculate the magnetic field at different points on the x-axis and find where it becomes zero.

1. Define a coordinate system: Let's assume the positive y-direction is pointing upwards, and the positive x-direction is pointing towards the right. The x-axis intersects the two wires at x = -2.0 cm and x = 2.0 cm.

2. Use the Biot-Savart Law: The Biot-Savart Law states that the magnetic field dB created by a small segment of wire carrying current is given by:

dB = (μ₀/4π) * Idl x r / r²

Here, μ₀ is the permeability of free space (constant), Idl represents the current element, r is the position vector from the element to the point where the magnetic field is being calculated, and x denotes the cross product.

3. Calculate the magnetic field at a point on the x-axis: Consider a point P on the x-axis located at a distance x from the wire carrying 5.0 A current. The magnetic field at P due to this wire can be calculated by integrating dB over the length of the wire. Let's call this B₁.

4. Repeat the same calculation for the other wire: Now consider the wire carrying 2.5 A current. Calculate the magnetic field at point P due to this wire, and denote it as B₂.

5. Find the point where the magnetic fields cancel out: Subtract the two magnetic fields, B₁ and B₂, and find the point on the x-axis where their difference is zero. This would be the location where the magnetic field is zero.

PART B:
If the currents in the two wires are in opposite directions, the magnetic fields created by them will be in opposite directions as well. In this case, the two magnetic fields will add up or cancel each other out based on their magnitudes.

Follow the same steps as in Part A to calculate the magnetic fields B₁ and B₂ at different points on the x-axis, taking into account the opposite direction of currents.

Find the point on the x-axis where the sum of the magnetic fields, B₁ + B₂, is zero. This would be the location where the magnetic field is zero if the currents are in opposite directions.

part a is 0.67cm