A long-straight wire carries a current and is aligned with the North-South direction. You place a compass 4 cm directly above the wire and above the North/East/South/West plane. The compass needle, which initially pointed North (far away from the wire) is now deflected by 20o from the North direction pointing West. (a)What is the direction of the conventional current? (b) What is the unit vector of the net magnetic field due to the wire and the Earth at the location of the compass? (c) If the Earth’s magnetic field magnitude is 0.05 mT in the plane of the compass and pointing North, then what is the current in the wire?

To answer these questions, we need to understand the relationship between a current-carrying wire and the magnetic field it produces.

(a) Direction of Conventional Current: Whenever a compass needle deflects to the west, it suggests that the magnetic field produced by the wire is oriented in the direction opposite to the compass needle's deflection. In this case, the compass needle deflects 20 degrees to the west, so the conventional current must be flowing in the opposite direction - to the east.

(b) Unit Vector of Net Magnetic Field: To determine the unit vector of the net magnetic field, we need to consider the contributions from both the wire and the Earth's magnetic field.

The wire produces a magnetic field that forms concentric circles around it when viewed from above. To find the direction, we can apply the right-hand rule: If you point your right thumb in the direction of the conventional current, your curled fingers will indicate the direction of the magnetic field. In this case, the magnetic field is pointing counterclockwise when viewed from above.

The Earth's magnetic field is given as 0.05 mT in the plane of the compass and pointing north. So, the unit vector of the Earth's magnetic field is in the north direction.

To find the net magnetic field, we need to sum the contributions of the wire and the Earth's magnetic field. Their vector sum will give us the net magnetic field at the location of the compass.

(c) Determining the Current in the Wire: To find the current in the wire, we can use the Biot-Savart law, which relates the magnetic field produced by a current-carrying wire to the current and distance.

The Biot-Savart law can be represented as:

B = (μ₀ / 4π) * (I / r)

Where:
B - Magnetic field
μ₀ - Permeability of free space (constant)
I - Current in the wire
r - Distance from the wire

Given that the compass is 4 cm (or 0.04 m) above the wire, we can write:

B_wire = (μ₀ / 4π) * (I / 0.04)

Now, the net magnetic field at the compass location is the vector sum of the wire's magnetic field and the Earth's magnetic field. Let's denote it as B_net.

The net magnetic field at the location of the compass is given by:

B_net = B_wire + B_earth

We can now substitute the known values into the equation and solve for the current (I):

B_net = B_wire + B_earth

Substituting B_wire and the given values:

B_net = (μ₀ / 4π) * (I / 0.04) + 0.05

Now, you can solve this equation for the current (I).