Question

Assume that a lightning bolt can be modeled as a long, straight line of current. If 15.0 C of charge passes by a point in \(1.50 \cdot 10^{-3}\) s, what is the magnitude of the magnetic field at a distance of \(26.0 \mathrm{~m}\) from the lightning bolt? a) \(7.69 \cdot 10^{-5} \mathrm{~T}\) c) \(4.21 \cdot 10^{-2} \mathrm{~T}\) e) \(2.22 \cdot 10^{2}\) T b) \(9.22 \cdot 10^{-3} \mathrm{~T}\) d) \(1.11 \cdot 10^{-1} \mathrm{~T}\)

Short answer

a) \(7.69\cdot10^{-5}\mathrm{T}\) b) \(5.98\cdot10^{-5}\mathrm{T}\) c) \(1.23\cdot10^{-3}\mathrm{T}\) d) \(1.43\cdot10^{-6}\mathrm{T}\) Solution: Step 1 - Calculate the current: \(I = \frac{15.0 C}{1.50 \cdot 10^{-3} s} = 10,000 A\) Step 2 - Use Ampere's Law to find the magnetic field: \(B = \frac{4\pi \cdot 10^{-7} Tm/A \cdot 10,000 A}{2\pi \cdot 26.0 m} = 7.69 \cdot 10^{-5} T\) Answer: The magnitude of the magnetic field at a distance of 26.0 m from the lightning bolt is \(7.69 \cdot 10^{-5} \mathrm{T}\), which corresponds to option (a).

Step-by-step solution

Calculate the current

The current, I, can be found using the formula \(I = \frac{Q}{t}\), where Q is the charge and t is the time. Plug in the given values of Q = 15 C and t = \(1.50 \cdot 10^{-3}\) s to find the current: \(I = \frac{15.0 C}{1.50 \cdot 10^{-3} s} = 10,000 A\)

Use the Biot-Savart Law (or Ampere's Law) to find the magnetic field

The magnetic field around a straight current-carrying wire can be calculated using the Biot-Savart Law. However, for a long, straight wire, it's simpler to use Ampere's Law, which states that \(B = \frac{\mu_0 I}{2\pi r}\), where \(B\) is the magnetic field, \(\mu_0\) is the vacuum permeability constant (\(4 \pi \cdot 10^{-7} Tm/A\)), \(I\) is the current, and \(r\) is the distance from the wire. Plug in the values for the current I = 10,000 A, distance r = 26.0 m, and \(\mu_0 = 4 \pi \cdot 10^{-7} Tm/A\): \(B = \frac{4\pi \cdot 10^{-7} Tm/A \cdot 10,000 A}{2\pi \cdot 26.0 m} = \frac{4 \cdot 10^{-3} T}{2 \cdot 26.0 m} = 7.69 \cdot 10^{-5} T\)

Identify the answer

The magnitude of the magnetic field at a distance of 26.0 m from the lightning bolt is \(7.69 \cdot 10^{-5} \mathrm{T}\). Thus, the correct answer is option (a).