Problem 16 - University Physics with Modern Physics

Question

A very long, straight horizontal wire carries a current such that \(3.50 \times 10^{18}\) electrons per second pass any given point going from west to east. What are the magnitude and direction of the magnetic field this wire produces at a point 4.00 \(\mathrm{cm}\) directly above it?

Step-by-Step Solution

Answer

The magnetic field is \(2.8 \times 10^{-6}\) T pointing north.

Step 1: Understand the Problem

We are tasked with finding the magnetic field produced by a long, straight current-carrying wire at a point above it. Given the number of electrons passing a point per second, we'll first convert this to current, then use it to find the magnetic field.

Step 2: Convert Electron Flow to Current

The number of electrons per second is given as \(3.50 \times 10^{18}\). Since each electron has a charge of \(1.6 \times 10^{-19}\) C, the current \(I\) is calculated by \(I = 3.50 \times 10^{18} \times 1.6 \times 10^{-19} = 0.56\) A, flowing west to east.

Step 3: Use Biot-Savart Law for Magnitude

For a long, straight wire, the magnetic field at a distance \(r\) is \(B = \frac{\mu_0 I}{2 \pi r}\). With \(\mu_0 = 4\pi \times 10^{-7}\, \text{T}\cdot\text{m/A}\) and \(r = 0.04\) m, substitute these into the equation to get \(B = \frac{(4\pi \times 10^{-7}) \times 0.56}{2 \pi \times 0.04} = 2.8 \times 10^{-6}\) T.

Step 4: Determine Direction Using Right-Hand Rule

By the right-hand rule, point your thumb in the direction of the current (west to east). Your fingers curl in the direction of the magnetic field. At a point above the wire, this results in the magnetic field pointing north.

Key Concepts

Biot-Savart Law

The Biot-Savart Law is essential for understanding how magnetic fields originate from moving charges or electric currents. This law gives us the mathematical way to calculate the magnetic field produced at a specific point in space by a current-carrying conductor. It's particularly useful when dealing with geometry and configurations where Ampere’s Law becomes challenging to apply.

In essence, the Biot-Savart Law states that the magnetic field, denoted as \( B \), at a point is proportional to the current \( I \) and depends inversely on the distance \( r \) from the current element. For an infinitely straight conductor, the calculation simplifies to the formula:
\[B = \frac{\mu_0 I}{2\pi r}\]
where \(\mu_0\) is the permeability of free space, valued at \(4\pi \times 10^{-7} \, \text{T}\cdot\text{m/A}\). This formula is useful for finding the magnetic field at any point around a long, straight wire, like the example from the exercise.

• The direct proportionality to current means stronger currents produce stronger fields.
• The inversely proportional relationship with distance reminds us that moving further from the wire decreases the magnetic field strength.

Right-Hand Rule

The Right-Hand Rule is a handy mnemonic for understanding the direction of the magnetic field relative to the current in a wire. It simplifies the potentially confusing task of determining magnetic field directions in three dimensions.

To apply the right-hand rule, you use your right hand to mimic the orientation of the current and the resultant magnetic field.

• Extend your thumb in the direction of the current flow. In the scenario from the exercise, this means pointing your thumb from west to east.
• Your fingers then naturally curl around in the direction the magnetic field lines circle the wire.
• Above the wire, your fingers point north, indicating that the magnetic field at that point wraps around the wire upwards towards the north.

The right-hand rule is intuitive once practiced, making it a powerful tool for visualizing and verifying field directions without calculations.

Current in a Wire

Current represents the flow of electric charge, and in our exercise, this flow is quantified by the number of electrons per second. Electron flow presents its behavior slightly differently from the conventional current since in reality, electrons move in the opposite direction of the conventional current.

In the given exercise, we compute the current \( I \) by multiplying the given electron flow rate by the charge per electron \( e \). This transformation to current (measured in amperes, A) becomes:
\[I = (3.50 \times 10^{18} \text{ electrons/s}) \times (1.6 \times 10^{-19} \text{ C/electron}) = 0.56 \text{ A}\]
This transformation helps us understand how much charge is moving past a certain point in the wire per second, highlighting the nature of current as a rate of charge flow.

• With current established, the problem can then utilize this value to determine the magnetic field produced.
• Understanding that the current flows west to east guides how we interpret and apply the right-hand rule for magnetic field direction.