Problem 15 - University Physics with Modern Physics

Question

The Magnetic Field from a Lightning Bolt. Lightning bolts can carry currents up to approximately 20 kA. We can model such a current as the equivalent of a very long, straight wire. (a) If you were unfortunate enough to be 5.0 \(\mathrm{m}\) away from such a lightning bolt, how large a magnetic field would you experience? (b) How does this field compare to one you would experience by being 5.0 \(\mathrm{cm}\) from a long, straight household current of 10 \(\mathrm{A} ?\)

Step-by-Step Solution

Answer

The magnetic field at 5.0 m from a lightning bolt is 0.8 mT. At 5.0 cm from a household wire, it's 40 µT. The lightning bolt's field is much larger.

Step 1: Understanding the Problem

We have a current-carrying conductor (equivalent to a very long, straight wire), and we want to calculate the magnetic field it produces at a certain distance. For Section (a), the lightning bolt is 5.0 m away and carries 20,000 A (20 kA). For Section (b), a household current of 10 A is 0.05 m away.

Step 2: Using the Magnetic Field Formula for a Straight Wire

The magnetic field caused by a long, straight wire at a distance can be calculated using the formula: \[ B = \frac{\mu_0 I}{2\pi r} \]where \(B\) is the magnetic field in teslas (T), \(\mu_0 = 4\pi \times 10^{-7} \text{T}m/ ext{A}\) is the permeability of free space, \(I\) is the current, and \(r\) is the distance from the wire.

Step 3: Calculating the Magnetic Field for the Lightning Bolt

Substitute the given values into the formula for part (a): Current \(I = 20000\,\text{A}\) and distance \(r = 5\,\text{m}\). Thus, \[ B_\text{lightning} = \frac{(4\pi \times 10^{-7} ) \times 20000}{2\pi \times 5} \]Calculate to find that \[ B_\text{lightning} = 8\times 10^{-4} \text{T} \] or 0.8 mT.

Step 4: Calculating the Magnetic Field for the Household Current

For the household current in part (b), with \(I = 10\,\text{A}\) and \(r = 0.05\,\text{m}\), substitute the values:\[ B_\text{household} = \frac{(4\pi \times 10^{-7} ) \times 10}{2\pi \times 0.05} \]Calculate to find that \[ B_\text{household} = 4\times 10^{-5} \text{T} \] or 40 µT.

Step 5: Comparing the Two Magnetic Fields

Now compare the two fields calculated in parts (a) and (b): - Lightning bolt magnetic field: 0.8 mT - Household current magnetic field: 40 µT The magnetic field from the lightning bolt is significantly larger than the field from the household current at the given distances.

Key Concepts

Lightning Bolt Current

A lightning bolt is an immensely powerful natural phenomenon that can carry a tremendous amount of electric current. When we say that a lightning bolt can carry up to 20,000 amperes (A) or 20 kiloamperes (kA), we are talking about a huge electric flow compared to everyday electrical appliances. To put things in perspective:

• A household appliance like a blender might use about 2-3 A.
• A toaster could use around 10 A.
• Thus, a lightning bolt's current is several thousand times stronger than these devices.

This vast amount of current can generate a substantial magnetic field, which can influence objects even at a distance like 5 meters away. This makes understanding the magnetic effects generated by such high currents crucial for our safety during storms.

Straight Wire Magnetic Field Formula

The magnetic field that surrounds a current-carrying wire is a fundamental concept in electromagnetism. For a very long, straight wire, this field is given by the formula: \[B = \frac{\mu_0 I}{2\pi r}\]where:

• \( B \) is the magnetic field measured in teslas (T),
• \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \) Tm/A,
• \( I \) is the current in amperes (A), and
• \( r \) is the distance from the wire in meters (m).

This formula helps calculate the magnetic field strength at a certain distance from the wire. It shows that the magnetic field is directly proportional to the current and inversely proportional to the distance from the wire. This inverse relationship means that as you move further away from the wire, the magnetic field decreases.

Magnetic Permeability

Magnetic permeability is a measure of how easily a material or space can support the formation of a magnetic field. It is a constant of nature in the vacuum, known as the permeability of free space (\( \mu_0 \)). In calculations for magnetic fields from wires, the value of \( \mu_0 \) is crucial, as it influences how strong the magnetic field will be for a given current and distance. The magnetic permeability tells us:

• How well a medium can "conduct" the magnetic field.
• It affects how magnetic fields interact with materials, influencing phenomena like magnetic induction and magnetism.

This constant value underlines the laws of electromagnetism and is essential for accurate calculations determining magnetic field strengths in practice.

Comparison of Magnetic Fields

The comparison of magnetic fields generated by different sources can reveal significant contrasts in magnetic field strengths. In the given exercise:

• The magnetic field from a lightning bolt at 5 m is calculated to be 0.8 milliteslas (mT).
• In contrast, the magnetic field from a typical household current at just 0.05 m is 40 microteslas (µT).

These values highlight several important points:

• A lightning bolt, even from a distance, produces a much stronger magnetic field than the household current up close.
• The substantially higher current in a lightning bolt contributes to its powerful magnetic field, despite both distances involved.
• This disparity in magnetic field strength emphasizes the sheer power of natural electrical phenomena compared to everyday household currents.

Understanding these comparisons helps underscore safety considerations around electrical currents, whether from natural or artificial sources.