How Does a Nearby Current Affect Compass Orientation?

[vasume]

Homework Statement

A compass needle points 20 degrees E of N outdoors. However, when it is placed 8.0 cm to the east of a vertical wire inside a building, it points 55 degrees E of N. What is the magnitude and direction of the current in the wire?
The Earth's field there is 0.50 x 10^ -4 T and is horizontal.

Homework Equations

I'm thinking I'll need to use B = ( u I ) / (2 r pi) and something involving vectors...

The Attempt at a Solution

I really don't even know where to begin with the setup for problem. Do I start with the angles to find a vector sum of something? I just have no idea, sorry. Someone please help walk me through it.

 

[anathema]

Hello!

Firstly, let's start by defining what we know and what we are trying to find. We know that the compass needle is pointing at a different angle when it is placed near the wire, and we are trying to find the magnitude and direction of the current in the wire.

We can start by drawing a diagram to visualize the situation. Draw a horizontal line to represent the ground, and label it as the x-axis. Then, draw a vertical line to represent the wire, and label it as the y-axis. The compass needle is pointing 20 degrees E of N outdoors, so we can draw a line at a 20 degree angle from the north direction on the x-axis. This represents the direction of the Earth's magnetic field.

Next, we need to consider the effect of the wire on the compass needle. When the compass is placed 8.0 cm to the east of the wire, it points 55 degrees E of N. This means that the wire is affecting the direction of the compass needle by 35 degrees (55-20=35). We can draw a line at a 35 degree angle from the north direction on the x-axis to represent the new direction of the compass needle.

Now, let's consider the equations that we can use to solve this problem. We can use the equation B = (uI) / (2rpi), where B is the magnetic field, u is the permeability of free space (4pi x 10^-7 Tm/A), I is the current in the wire, and r is the distance from the wire. We also know that the Earth's field is 0.50 x 10^-4 T and is horizontal, so we can use this information to find the magnitude and direction of the current in the wire.

To find the magnitude of the current, we can rearrange the equation to solve for I. This gives us I = (2rpiB) / u. We know that r = 8.0 cm = 0.08 m, and B = 0.50 x 10^-4 T, so we can plug these values into the equation to find the current. I = (2 x 0.08 x pi x 0.50 x 10^-4) / (4pi x 10^-7) = 0.4 A.

To find the direction of the current, we can use the right hand rule

 

[m2003]

I would approach this problem by first identifying the key concepts involved: magnetism, currents, and the Earth's magnetic field. From there, I would use the given information to set up a mathematical equation that relates these concepts. In this case, the equation you mentioned, B = (u I) / (2 r pi), is a good starting point. This equation represents the magnetic field (B) created by a current (I) at a distance (r) from the wire, with u being a constant known as the permeability of free space.

Next, I would use the given angles to determine the direction of the current. Since the compass needle points to the east of north in both cases, we can assume that the current is flowing in a direction perpendicular to the compass needle, which in this case would be towards the south.

Using the given distance of 8.0 cm and the Earth's magnetic field of 0.50 x 10^-4 T, we can plug these values into the equation B = (u I) / (2 r pi) and solve for the current, I. This will give us the magnitude of the current in the wire.

Finally, to determine the direction of the current, we can use the right-hand rule, which states that if you point your thumb in the direction of the current, your fingers will curl in the direction of the magnetic field. In this case, since the compass needle points east of north, the magnetic field must be pointing towards the east. Therefore, using the right-hand rule, we can determine that the current is flowing south to north in the wire.

In summary, to solve this problem, we need to use the equation B = (u I) / (2 r pi) to calculate the magnitude of the current, and then use the right-hand rule to determine the direction of the current. It is also important to keep in mind the key concepts of magnetism, currents, and the Earth's magnetic field to properly interpret the given information.