Problem 16 - University Physics with Modern Physics

Question

A very long, straight horizontal wire carries a current such that 8.20 \(\times\) 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 cm directly above it?

Step-by-Step Solution

Answer

The magnetic field is \(6.56 \times 10^{-6}\) T, directed into the page.

Step 1: Calculate the Current

To find the current, we use the formula \(I = n \cdot e\), where \(n\) is the number of electrons passing per second and \(e\) is the charge of an electron (\(1.6 \times 10^{-19}\) C). Thus, \(I = 8.20 \times 10^{18} \times 1.6 \times 10^{-19} = 1.312\) A.

Step 2: Convert Distance to Meters

The distance from the wire to the point is 4.00 cm. Convert this to meters: \(4.00\) cm = \(0.040\) m.

Step 3: Apply the Biot-Savart Law

For a long straight wire, the magnetic field at a distance \(r\) from the wire is given by \(B = \frac{\mu_0 I}{2\pi r}\). We know \(\mu_0 = 4\pi \times 10^{-7}\) T\(\cdot\)m/A.

Step 4: Calculate the Magnetic Field

Substitute the values into the formula: \(B = \frac{4\pi \times 10^{-7} \cdot 1.312}{2\pi \cdot 0.040}\). Simplifying this gives \(B = \frac{4 \times 10^{-7} \cdot 1.312}{0.080}\) = \(6.56 \times 10^{-6}\) T.

Step 5: Determine the Direction Using Right-Hand Rule

Using the right-hand rule, point your thumb in the direction of the current (west to east). Your curled fingers indicate the direction of the magnetic field, which will be into the page at the point 4 cm above the wire.

Key Concepts

Current in Wire

When we talk about the current in a wire, we're discussing the flow of electric charge, often carried by electrons. In this context, a current is measured by the number of electrons passing through a point in the wire per second. We calculate current using the formula:

• \(I = n \times e\)

where \(I\) is the current, \(n\) is the number of electrons per second, and \(e\) is the charge of an electron \( (1.6 \times 10^{-19} \text{ C})\).
This tells us how many charges flow through the wire, representing an electric current in amperes (A). For instance, if 8.20 \(\times\) 10\(^{18}\) electrons per second pass through the wire, it means that the current is 1.312 A.
This flow of electrons forms the backbone of many modern technologies, providing energy to power circuits and devices.

Biot-Savart Law

The Biot-Savart Law is fundamental in understanding how magnetic fields are generated by electric currents. It's a powerful tool used to calculate the magnetic field

• generated by a current-carrying wire.
• The magnetic field at a point in space is determined by the product of the current and a constant.

The formula is given by \[B = \frac{\mu_0 I}{2\pi r}\],
where \(B\) is the magnetic field, \(\mu_0\) is the permeability of free space \((4\pi \times 10^{-7} \text{ T}\cdot\text{m/A})\), \(I\) is the current, and \(r\) is the distance from the wire.
This law is crucial for engineers and physicists for designing and understanding circuits, motors, and any device involving magnetism.

Right-Hand Rule

The right-hand rule is a simple yet effective method to determine the direction of the magnetic field around current-carrying wires. This rule helps visualize how the magnetic field "flows" around the wire.

• Point your right thumb in the direction of the current.
• Curl your fingers around the wire.

Now, your fingers indicate the circular direction of the magnetic field lines.
In this exercise, with the current flowing from west to east, your thumb points east, and your fingers curl into the page above the wire.
This is a great tool for remembering the orientation of magnetic fields in various physical setups.

Electron Flow

Understanding electron flow is key to mastering electrical circuits. Electrons, being negatively charged, flow opposite to conventional current direction. In most scenarios, we consider current to flow from positive to negative terminal.

• In the exercise, electrons moving from west to east means the conventional current goes from east to west.

This distinction between electron flow and current flow is essential to accurately describe electrical phenomena.
Considering both, we can understand the full behavior of circuits and the resulting magnetic effects.

Magnetic Field Direction

The direction of the magnetic field is a crucial aspect when calculating its effects and understanding its impact within electrical systems. The magnetic effect caused by currents creates a field perpendicular to the current's direction.

• In this scenario, with the wire's current running west to east, the right-hand rule shows the magnetic field directed into the page above the wire.

Magnetic fields will always orient themselves in a direction that loops around the path of the current.
This directionality is important for designing devices and ensuring their proper functioning in technologies such as electromagnets and transformers.