Question

You are standing at a spot where the magnetic field of the Earth is horizontal, points due northward, and has magnitude \(40.0 \mu \mathrm{T}\). Directly above your head, at a height of \(12.0 \mathrm{~m},\) a long, horizontal cable carries a steady \(\mathrm{DC}\) current of 500.0 A due northward. Calculate the angle \(\theta\) by which your magnetic compass needle is deflected from true magnetic north by the effect of the cable. Don't forget the sign of \(\theta-\) is the deflection eastward or westward?

Short answer

Given: Magnetic field of the Earth: \(40.0 \, \mu T\) Height of the cable above the ground: \(12.0 \, m\) Current carried by the cable: \(500.0 A\) Following the steps outlined in the solution, carry out the calculations and find the angle of deflection, which is: \(\theta = -22.7°\) The compass needle is deflected by an angle of \(-22.7°\) westward from the true magnetic north due to the effect of the horizontal cable carrying the steady DC current.

Step-by-step solution

Calculate the magnetic field due to the wire

First, we need to calculate the magnetic field (\(B_{wire}\)) produced by the wire at the position of the person using the formula for the magnetic field produced by a long straight current-carrying wire. The formula is given by: \(B_{wire} = \frac{\mu_0 I}{2\pi d}\) Where \(B_{wire}\) is the magnetic field produced by the wire, \(\mu_0\) is the permeability of free space (\(4\pi \times 10^{-7} \, Tm/A\)), \(I\) is the current in the wire (500.0 A), and \(d\) is the distance from the wire to the person (12.0 m). Now, let's plug in the values and calculate the magnetic field due to the wire. \(B_{wire} = \frac{4\pi \times 10^{-7} \, Tm/A * 500.0 \, A}{2\pi * 12.0 \, m} = 1.67 \times 10^{-5} \, T\)

Find the net magnetic field

Next, let's find the net magnetic field (\(B_{net}\)) by adding the Earth's magnetic field (\(B_{earth}\)) which is given as \(40.0 \, \mu T\) (horizontal and northward) and the magnetic field due to the wire (\(B_{wire}\)), which is horizontal and points westward. Since the Earth's magnetic field and the magnetic field due to the wire are perpendicular to each other, we can use the Pythagorean theorem to find the magnitude of the net magnetic field. \(B_{net} = \sqrt{B_{earth}^2 + B_{wire}^2} = \sqrt{(40.0 \times 10^{-6}\, T)^2 + (1.67 \times 10^{-5}\, T)^2} = 4.32 \times 10^{-5} \, T\)

Calculate the angle of deflection

Now that we have the net magnetic field, we can find the angle \(\theta\) of deflection by considering the ratio of the magnetic field due to the wire (\(B_{wire}\)) and the Earth's magnetic field (\(B_{earth}\)). We can use the tangent function and calculate the angle as follows: \(\tan{\theta} = \frac{B_{wire}}{B_{earth}}\) \(\theta = \arctan{\frac{B_{wire}}{B_{earth}}} = \arctan{\frac{1.67 \times 10^{-5}\, T}{40.0 \times 10^{-6} \, T}} = 22.7°\) Since the magnetic field due to the wire points westward, the compass needle is deflected by \(22.7°\) westward (which is negative in this context). So, \(\theta = -22.7°\).

Key concepts

Magnetic Field

Magnetic fields are invisible forces that permeate space and affect magnetic materials and charged particles. They are produced by electric currents, such as those that flow through a wire or cable, as well as by certain natural phenomena like Earth's magnetic field. A magnetic field can be described using field lines or vectors that exert force on objects within the field, causing them to move or align.

The strength and direction of a magnetic field at a point in space are characterized by a vector quantity known as the magnetic induction or magnetic flux density, often denoted by the symbol \(B\), and measured in Tesla (T).

• A magnetic field can be thought of as a set of vector quantities that describe the magnetic influence on a certain region.
• The magnetic field influences compass needles, causing them to align with the field direction.

Understanding magnetic fields is crucial as they significantly impact how many devices operate, ranging from compasses to large-scale electrical systems.

Current-Carrying Wire

An electric current passing through a wire generates a magnetic field that surrounds the wire. This forms the basis for many electrical devices and phenomena, such as electromagnets and transformers. The concept is described by Ampere’s Law, which states that magnetic fields are generated by moving charges or currents.

For a long straight current-carrying wire, the magnetic field produced is circular around the wire and diminishes with distance. The formula for the magnetic field generated by a long, straight wire is given by:
\[B_{wire} = \frac{\mu_0 I}{2\pi d}\]
where \(\mu_0\) is the permeability of free space, \(I\) is the current, and \(d\) is the distance from the wire.

• The direction of the magnetic field follows the right-hand rule: if you wrap your right hand around the wire with your thumb pointing in the current direction, your fingers will curl in the direction of the magnetic field.
• This principle allows us to calculate the precise effect a current has on nearby conductive materials.

Such understanding is critical in designing and interpreting experiments and devices involving electric currents.

Earth's Magnetic Field

The Earth acts like a giant magnet with its magnetic field extending from its core into space. This geomagnetic field shields the planet from solar wind and cosmic radiation, making it essential for life on Earth. The field is largely horizontal at most locations on the Earth's surface, aligning generally from the magnetic south to the magnetic north.

This magnetic field is why compasses work, as they align themselves to point towards the magnetic north. The Earth's magnetic field is not uniform and changes over time due to the dynamic processes in the Earth's outer core. Typically, its strength at the surface ranges from about 25 to 65 microteslas \(\mu T\).

• Geomagnetic field lines exit the Earth near the magnetic south pole and re-enter near the magnetic north pole.
• The field's magnitude and direction can vary, influencing navigation and affecting satellite and communication systems.

Understanding the Earth's magnetic field facilitates navigation and offers insight into geophysical processes.

Angle of Deflection

The angle of deflection refers to the angle that a compass needle deviates from true North due to the presence of an external magnetic field. In contexts like the above exercise, this occurs when a current-carrying wire generates a magnetic field that interacts with the Earth's magnetic field, causing compasses to deflect.

The deflection is calculated using basic trigonometry, which considers the perpendicular component of the wire's magnetic field to the Earth's magnetic field. The formula for calculating the angle of deflection \(\theta\) is:
\[\theta = \arctan\left(\frac{B_{wire}}{B_{earth}}\right)\]
Where \(B_{wire}\) is the magnetic field due to the wire, and \(B_{earth}\) is the Earth's magnetic field. The direction of deflection is determined by the direction of the external magnetic field generated by the current.

• Positive deflection angles are typically denoted by displacement eastward, whereas negative angles signify westward deflection.
• Such calculations are vital in understanding and predicting the effects of magnetic fields on navigation and instrumentation.

Angle of deflection provides key insights into how additional magnetic influences can alter the behavior of instruments like compasses.