Problem 17 - 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 \(\mathrm{cm}\) 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) 10 A; (b) Above the wire; (c) East or West of the wire.

Step 1: Determine the Formula

For a long, straight current-carrying wire, the magnetic field at a distance \( r \) from the wire is given by the formula \( B = \frac{\mu_0 I}{2\pi r} \), where \( B \) is the magnetic field, \( I \) is the current, and \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \mathrm{~T \cdot m/A} \)).

Step 2: Rearrange for Current

To find the current \( I \) when the magnetic field \( B = 1.00 \times 10^{-4} \mathrm{~T} \) (1 Gauss) at a distance \( r = 0.02 \mathrm{~m} \), rearrange the formula: \( I = \frac{2\pi r B}{\mu_0} \).

Step 3: Calculate the Current

Substitute \( r = 0.02 \mathrm{~m} \), \( B = 1.00 \times 10^{-4} \mathrm{~T} \), and \( \mu_0 = 4\pi \times 10^{-7} \mathrm{~T \cdot m/A} \) into the formula: \[ I = \frac{2 \pi \times 0.02 \text{ m} \times 1.00 \times 10^{-4} \text{ T}}{4\pi \times 10^{-7} \text{ T m/A}} = 10 \text{ A} \].

Step 4: Understand Direction of Magnetic Field

For a wire, the magnetic field direction follows the right-hand rule: if the thumb points in the direction of the current, the fingers curl in the direction of the magnetic field.

Step 5: Determine Locations for Horizontal Wire

If the wire is horizontal and current flows east to west, the magnetic field forms concentric circles in a plane perpendicular to the wire. The field above the wire points north, and below the wire, it points south.

Step 6: Determine Locations for Vertical Wire

If the wire is vertical and current flows upward, the magnetic field circulates in horizontal circles. East and west of the wire, the field points north; north and south of the wire, the field points east and west, respectively.

Key Concepts

Right-hand Rule

The right-hand rule is a simple yet effective technique used to determine the direction of the magnetic field around a current-carrying wire. To use the right-hand rule, follow these steps:

• Imagine grasping the wire with your right hand.
• Point your thumb in the direction of the current.
• The curl of your fingers will show you the direction of the magnetic field.

When the current flows through the wire, the magnetic field generated is circular around the wire. This means that no matter where you stand around the wire, the direction of the magnetic field can be determined using this rule. This simple hand gesture helps you visualize the three-dimensional shape of the magnetic field, making it easier to understand complex arrangements of wires and currents.
This technique is especially useful when working with wires set in various arrangements, enabling a quick determination of the magnetic field's orientation.

Permeability of Free Space

The permeability of free space, denoted by \(\mu_0\), is a fundamental constant in physics that describes how a magnetic field influences and propagates through a vacuum. It is crucial in the calculation of the magnetic field around a current-carrying wire. The value of \(\mu_0\) is \(4\pi \times 10^{-7} \, \text{T \cdot m/A}\). This constant appears in the formula for calculating the magnetic field of a long, straight wire:

• \(B = \frac{\mu_0 I}{2\pi r}\)

Here, \(B\) represents the magnetic field at a distance \(r\) from the wire, and \(I\) is the current running through the wire.
Understanding this constant is vital for any calculations involving magnetic fields, as it ensures that the computed values are accurate and reliable. Without \(\mu_0\), our ability to analyze and predict how magnetic fields behave in space would be considerably limited.

Magnetic Field Direction

The direction of the magnetic field around a current-carrying wire is often visualized using concentric circles in a plane surrounding the wire. The orientation of these circles depends on the wire's positioning and the current flow.

• For a horizontal wire with current running from east to west, the magnetic field above the wire points north, while it points south below.
• For a vertical wire with upward current, the field circulates horizontally. East and west of the wire, it directs north. Conversely, north and south of the wire, the field flows east and west, respectively.

This systematic patterning of the magnetic field can easily be understood using diagrams and physical rules, such as the right-hand rule. Visualizing these patterns helps in practical applications, such as in designing circuits or studying electromagnetic effects in different scenarios.