Question

A supersonic aircraft with a wingspan of $10.0\mathrm{m}$ is flying over the north magnetic pole (in a magnetic field of magnitude 0.500 G perpendicular to the ground) at a speed of three times the speed of sound (Mach 3). What is the potential difference between the tips of the wings? Assume that the wings are made of aluminum.

Short answer

Answer: The potential difference between the tips of the wings is approximately 5.14 mV.

Step-by-step solution

Convert the magnetic field strength from G to Tesla

The given magnetic field strength is 0.500 G. We need to convert it into Tesla (T) before substituting it into the EMF formula. Since 1 G = 10^-4 T, we find: $B=0.500\mathrm{G}\times {10}^{-4}\frac{\mathrm{T}}{\mathrm{G}}=5.0\times {10}^{-5}\mathrm{T}$

Find the speed of sound in air

Standard speed of sound in air is approximately 343 m/s. Since the aircraft is flying at Mach 3, its actual speed is three times this value: $v=3\times 343\frac{\mathrm{m}}{\mathrm{s}}=1029\frac{\mathrm{m}}{\mathrm{s}}$

Calculate the electromotive force (EMF)

Now that we have all values needed, we can calculate the potential difference between the tips of the wings using the formula for EMF: EMF = B * L * v EMF = $(5.0\times {10}^{-5}\mathrm{T})\times (10.0\mathrm{m})\times (1029\frac{\mathrm{m}}{\mathrm{s}})$ EMF = $5.1415\times {10}^{-3}\mathrm{V}$

Round the answer

Finally, let's round the answer to two decimal places: EMF = $5.14\times {10}^{-3}\mathrm{V}$ Therefore, the potential difference between the tips of the wings is approximately 5.14 mV.

Key concepts

Supersonic Aircraft

Supersonic aircraft are fascinating flying machines that travel faster than the speed of sound. These jets move at speeds exceeding Mach 1, where Mach refers to the ratio of the speed of the object to the speed of sound in the surrounding medium.
For our exercise, the aircraft is flying at Mach 3, which means it travels three times faster than sound. This level of speed causes significant engineering and physical challenges, such as reducing drag and dealing with the sonic boom effect.
The aircraft's wings interact with the Earth's magnetic field as they cut through the air, generating a potential difference across the wingtips. Understanding this interaction involves grasping the basics of electromagnetic induction.

Magnetic Field Conversion

In the context of electromagnetic induction, converting magnetic field measurements is important. Magnetic field strength is often given in gauss (G), but for calculations like ours, we need it in teslas (T).

• 1 gauss (G) equals ${10}^{-4}$ tesla (T).

Thus, for a magnetic field strength of 0.500 G, the conversion to tesla goes as follows:

• $B=0.500\mathrm{G}\times {10}^{-4}\mathrm{T}/\mathrm{G}=5.0\times {10}^{-5}\mathrm{T}$

This ensures that when we use the EMF formula, our units are consistent, leading to accurate results.

Electromotive Force (EMF) Calculation

The calculation of electromotive force (EMF) is central to determining the potential difference created across the wings of a supersonic aircraft. This is a direct application of Faraday's Law of Electromagnetic Induction.

• EMF is calculated using the formula: $\text{EMF}=B\times L\times v$
• Where:
  • $B$ is the magnetic field (in teslas),
  • $L$ is the wingspan (in meters),
  • $v$ is the velocity of the aircraft (in meters per second).

For our scenario:

• $B=5.0\times {10}^{-5}\mathrm{T}$
• $L=10.0\mathrm{m}$
• $v=1029\mathrm{m}/\mathrm{s}$

Plugging these values into the formula gives us:

• EMF = $(5.0\times {10}^{-5}\mathrm{T})\times (10.0\mathrm{m})\times (1029\mathrm{m}/\mathrm{s})=5.1415\times {10}^{-3}\mathrm{V}$

Rounding to two decimal places, the EMF is approximately 5.14 mV, indicating a small but measurable voltage difference between the wingtips.

Speed of Sound

Understanding the speed of sound is critical when discussing supersonic travel. The speed of sound in air is roughly 343 m/s under normal conditions.

• This is referred to as Mach 1.

When an object, like our aircraft, travels at Mach 3, its speed is three times the speed of sound, so 343 m/s multiplied by three. Thus, the aircraft's speed is 1029 m/s.
This speed plays a pivotal role in our EMF calculations, as it directly influences how the wings cut through the magnetic field, generating a potential difference.