Problem 40 - College Physics

Question

You want to produce a magnetic field of magnitude \(5.50 \times 10^{-4} \mathrm{T}\) at a distance of 0.040 \(\mathrm{m}\) from a long, straight wire's center. (a) What current is required to produce this field? (b) With the current found in part (a), how strong is the magnetic field 8.00 \(\mathrm{cm}\) from the wire's center?

Step-by-Step Solution

Answer

(a) 11.0 A; (b) 2.75 × 10^-4 T.

Step 1: Understanding the formula

The magnetic field around a long, straight wire is given by the formula: \( B = \frac{\mu_0 I}{2\pi r} \), where \( B \) is the magnetic field, \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \ \mathrm{T\cdot m/A} \)), \( I \) is the current, and \( r \) is the distance from the wire. We will use this formula to find the current \( I \).

Step 2: Rearranging the formula

To find the current \( I \), we rearrange the formula to solve for \( I \): \( I = \frac{B \, 2\pi r}{\mu_0} \).

Step 3: Substituting the values for part (a)

Substitute the given values into the rearranged formula: \( B = 5.50 \times 10^{-4} \, \mathrm{T} \), \( r = 0.040 \, \mathrm{m} \), and \( \mu_0 = 4\pi \times 10^{-7} \, \mathrm{T\cdot m/A} \). This gives \( I = \frac{(5.50 \times 10^{-4} \, \mathrm{T}) \, 2\pi \, (0.040 \, \mathrm{m})}{4\pi \times 10^{-7} \, \mathrm{T\cdot m/A}} \).

Step 4: Calculating the current

Perform the calculation: \( I = \frac{(5.50 \times 10^{-4} \, \mathrm{T}) \, 2\pi \, (0.040 \, \mathrm{m})}{4\pi \times 10^{-7} \, \mathrm{T\cdot m/A}} = 11.0 \, \mathrm{A} \). So, the current required is 11.0 A.

Step 5: Calculating the magnetic field at 8.00 cm

Now for part (b), we use the same formula \( B = \frac{\mu_0 I}{2\pi r} \) to find the magnetic field at \( r = 0.080 \, \mathrm{m} \) with the current \( I = 11.0 \, \mathrm{A} \). Substitute the values: \( B = \frac{(4\pi \times 10^{-7} \, \mathrm{T\cdot m/A}) \, (11.0 \, \mathrm{A})}{2\pi \, (0.080 \, \mathrm{m})} \).

Step 6: Solving for the magnetic field at 8.00 cm

Perform the calculation: \( B = \frac{(4\pi \times 10^{-7} \, \mathrm{T\cdot m/A}) \, (11.0 \, \mathrm{A})}{2\pi \, (0.080 \, \mathrm{m})} = 2.75 \times 10^{-4} \, \mathrm{T} \). Thus, the magnetic field at 8.00 cm is \( 2.75 \times 10^{-4} \, \mathrm{T} \).

Key Concepts

Ampere's Law

Ampere's Law is a fundamental principle in electromagnetism that describes how magnetic fields are generated by electric currents. It states that the line integral of the magnetic field \( \mathbf{B} \) around a closed loop is proportional to the total current \( I \) passing through the loop. Mathematically, this is expressed as: \[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I \]Here, \( \mu_0 \) represents the permeability of free space, and \( d\mathbf{l} \) is a differential length element of the loop. Ampere's Law simplifies to show that for a long, straight wire, the magnetic field's strength depends on the current, and diminishes with distance from the wire. This law is essential for calculating the magnetic effects of currents in various scenarios. By applying this principle, the relation between a wire's current and the magnetic field it generates can be understood and utilized in electric circuits and devices.

Current Calculation

The current calculation is important when trying to determine the amount of current needed to generate a specific magnetic field at a certain distance from a wire. In this problem, we use the formula that comes from Ampere's Law specific to a long, straight wire:\[ I = \frac{B \cdot 2\pi r}{\mu_0} \]This equation is derived by rearranging the magnetic field equation to solve for the current \( I \).

• \( B \) is the magnetic field's desired strength.
• \( r \) is the distance from the wire center to the point where \( B \) is measured.
• \( \mu_0 \) is the permeability of free space.

For example, if the magnetic field required is \( 5.50 \times 10^{-4} \ \mathrm{T} \) at \( 0.040 \ \mathrm{m} \) from the wire, you can substitute these values into the formula to find \( I \). This calculation is crucial for electronic design and experimentation, ensuring circuits function as intended.

Permeability of Free Space

The permeability of free space, often denoted as \( \mu_0 \), is a physical constant integral to electromagnetic theory. It characterizes how much magnetic field can be generated within a vacuum by a given amount of current. Mathematically, it is expressed as: \[ \mu_0 = 4\pi \times 10^{-7} \, \mathrm{T \cdot m/A} \]This constant indicates the extent to which a magnetic field can penetrate a vacuum and is a key component when calculating magnetic fields and forces. In our context, \( \mu_0 \) allows us to understand the interaction between magnetic fields and currents in wire calculations. It is particularly important in equations derived from Ampere's Law, as it provides the 'scaling factor' that relates the current to the magnetic field strength for any given geometry. Understanding \( \mu_0 \) is foundational to mastering how magnetic phenomena work in theoretical and applied physics.

Magnetic Field of a Straight Wire

The magnetic field produced by a long, straight wire is an intriguing phenomenon. When an electric current runs through a wire, it creates a circular magnetic field surrounding the wire. This field's strength decreases with distance from the wire. The relationship can be described with the formula: \[ B = \frac{\mu_0 I}{2\pi r} \]

• \( B \) is the magnetic field strength.
• \( \mu_0 \) is the permeability of free space.
• \( I \) is the current flowing through the wire.
• \( r \) is the radial distance from the wire's center.

For example, if we calculate \( B \) at 0.080 m from a wire with a current of 11.0 A, this formula shows how the magnetic field lessens with increasing distance. This principle has practical applications in designing electromagnetic devices and understanding how currents interact with their surroundings, serving as a basis for technologies in electronics and communications.