Problem 19 - University Physics with Modern Physics

Question

(a) How large a current would a very long, straight wire have to carry so that the magnetic field 2.00 cm from the wire is equal to 1.00 G (comparable to the earth's northward-pointing magnetic field)? (b) If the wire is horizontal with the current running from east to west, at what locations would the magnetic field of the wire point in the same direction as the horizontal component of the earth's magnetic field? (c) Repeat part (b) except the wire is vertical with the current going upward.

Step-by-Step Solution

Answer

(a) 1.00 A; (b) Above the wire (north); (c) East of the wire.

Step 1: Understand the Formula for Magnetic Field around a Wire

The magnetic field (\(B\) ) around a very long straight wire carrying current (\(I\) ) at a distance (\(r\) ) is given by Ampère's Law: \[B = \frac{\mu_0 I}{2\pi r}\]where \(\mu_0\) is the permeability of free space, \(\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A} \).

Step 2: Calculate the Current for Part (a)

We know \(B = 1.00 \text{ G} = 1.00 \times 10^{-4} \text{ T}\) (since 1 Gauss = \(10^{-4}\) Tesla) and \(r = 2.00 \text{ cm} = 0.02 \text{ m}\). Using the formula from Step 1:\[I = \frac{B \cdot 2\pi r}{\mu_0} = \frac{1.00 \times 10^{-4} \cdot 2\pi \cdot 0.02}{4\pi \times 10^{-7}}\]Simplifying, \[I = 1.00 \text{ A}\].

Step 3: Analyze the Direction for Part (b)

For a horizontal wire with current running east to west, apply the right-hand rule: point your thumb in the direction of the current (east to west) and your fingers curl in the direction of the magnetic field lines. The magnetic field above the plane of the wire points north, aligning with the earth’s magnetic field.

Step 4: Determine Locations for Magnetic Field Direction (Part b)

Since the wire runs east to west, the magnetic field above the wire will point north. Thus, locations above the wire (north of the wire) will have a magnetic field in the same direction as the earth's horizontal magnetic field component.

Step 5: Analyze for a Vertical Wire with Current Going Up (Part c)

For a vertical wire with current running upward, apply the right-hand rule again: point your thumb in the direction of the current (upwards) and your fingers curl around the wire. The magnetic field at a point to the east of the wire points north.

Step 6: Determine Magnetic Field Direction for Part (c)

With the wire vertical and current going upwards, the field direction to the east of the wire aligns with the earth's northward magnetic field. Thus, the magnetic field at points directly east of the wire is in the same direction as the earth's horizontal component.

Key Concepts

Ampère's Law

Ampère's Law is a fundamental principle that relates the magnetic field to the electric current producing it. Specifically, it states that the magnetic field around a closed loop is equal to the permeability of free space multiplied by the total current passing through the loop. It can be applied to calculate the magnetic field around a long straight wire carrying a current. When using Ampère's Law, the magnetic field (\(B\)) created by a current (\(I\)) flowing through a straight wire can be expressed as:

• \[ B = \frac{\mu_0 I}{2\pi r} \]

Where \(r\) is the radial distance from the wire, and \(\mu_0\) is the permeability of free space. This is particularly useful because it simplifies how we understand magnetic fields around wires, allowing us to accurately predict and calculate these fields for various practical applications.

Permeability of Free Space

The permeability of free space, often denoted as \(\mu_0\), is a constant that characterizes the ability of a vacuum to support the formation of a magnetic field. In mathematical terms, \(\mu_0\) is equal to \(4\pi \times 10^{-7} \text{ T}\cdot\text{m/A} \). This value is crucial when using Ampère's Law as it directly influences the strength of the magnetic field generated by the current.

Think of \(\mu_0\) as a bridge between electricity and magnetism. It provides a quantitative way to connect the number of amperes flowing through a conductor to the intensity of the magnetic field produced. In practice, \(\mu_0\) allows scientists and engineers to create models and calculations that are consistent across different environments, ensuring predictable behavior in electromagnetic applications.

Right-Hand Rule

The right-hand rule is a simple and practical tool used to determine the direction of the magnetic field relative to the current's direction. For a straight wire carrying a current, you can apply this rule by using your right hand:

• Point your thumb in the direction of the current.
• Let your fingers curl naturally.
• The direction in which your fingers curl is the direction of the magnetic field lines around the wire.

This method helps visualize the three-dimensional nature of magnetic fields. It is particularly useful when you have a wire positioned in a specific orientation and need to find out where the magnetic field will point.

Earth's Magnetic Field

Earth's magnetic field is similar to that of a giant bar magnet with a north and south pole. It extends from the planet's core to where it meets solar winds. The horizontal component of Earth's magnetic field plays a key role in navigation and is approximately equivalent to the strength of a typical magnetic field produced by a modest electric current in a wire positioned nearby.

For the exercise, we compare the magnetic field generated by the wire to the Earth's horizontal magnetic field. This context allows us to explore scenarios where a wire's field direction matches the Earth's magnetic field, making it easier to understand how different orientations and current directions affect the resulting magnetic field produced relative to the Earth's magnetic field.