Problem 49 - Physics

Question

In a lightning bolt, \(15 \mathrm{C}\) of charge flows during a time of \(1.5 \times 10^{-3} \mathrm{~s}\). Assuming that the lightning bolt can be represented as a long, straight line of current, what is the magnitude of the magnetic field at a distance of \(25 \mathrm{~m}\) from the bolt?

Step-by-Step Solution

Answer

The magnetic field at a distance of 25 m is \(8 \times 10^{-5} \mathrm{~T}\).

Step 1: Identify the Given Values

We are given the charge \( Q = 15 \mathrm{~C} \), the time \( t = 1.5 \times 10^{-3} \mathrm{~s} \), and the distance \( r = 25 \mathrm{~m} \).

Step 2: Calculate the Current

The current \( I \) flowing through the bolt can be found using the formula for current: \( I = \frac{Q}{t} \). We plug in the values: \[ I = \frac{15}{1.5 \times 10^{-3}} = 1 \times 10^4 \mathrm{~A} \].

Step 3: Use the Biot-Savart Law for Magnetic Field

The magnetic field \( B \) due to a long straight conductor is given by the formula: \[ B = \frac{\mu_0 I}{2 \pi r} \] where \( \mu_0 = 4\pi \times 10^{-7} \mathrm{~T}\cdot\mathrm{m/A} \).

Step 4: Plug in the Values

Substitute the known values into the formula: \[ B = \frac{(4\pi \times 10^{-7})(1 \times 10^4)}{2\pi (25)} \].

Step 5: Simplify the Expression

Calculate the magnetic field: \[ B = \frac{4 \times 10^{-3}}{50} = 8 \times 10^{-5} \mathrm{~T} \].

Key Concepts

Biot-Savart Law

The Biot-Savart Law is a fundamental principle in electromagnetism that allows us to calculate the magnetic field generated by a current-carrying conductor. This law is particularly useful when dealing with complex geometries or specific current distributions. In simple terms, it tells us how a tiny piece of current contributes to the total magnetic field at a certain point in space.

The law can be mathematically expressed as:\[B = \frac{\mu_0 I}{2 \pi r} \]where:

• \(B\) is the magnetic field.
• \(\mu_0\) is the permeability of free space, a constant \((4\pi \times 10^{-7} \text{ T}\cdot\text{m/A})\).
• \(I\) is the current flowing through the conductor.
• \(r\) is the perpendicular distance from the conductor to the point where the field is being calculated.

This formula is especially applicable to a long, straight wire, which simplifies calculations because the symmetry means the field points in a predictable direction and follows a circular path around the wire.

current calculation

Calculating current is a fundamental step in many physics problems, especially those involving flowing charges like a lightning bolt. Current is essentially the rate at which charge flows through a surface. It is a measure of how much charge passes through a given point per unit time.

The formula to calculate current \(I\) is:\[I = \frac{Q}{t}\]where:

• \(Q\) is the total charge in coulombs.
• \(t\) is the time in seconds over which the charge flows.

Applied to a lightning bolt, suppose we have \(15\) coulombs of charge flowing in \(1.5 \times 10^{-3}\) seconds. Plugging these values into the formula gives us:\[I = \frac{15}{1.5 \times 10^{-3}} = 1 \times 10^4 \text{ A}\]This result tells us that the lightning bolt carries an incredibly high current, highlighting the power and danger of such natural phenomena.

lightning bolt

A lightning bolt is a massive electrical discharge between clouds or from clouds to the ground. This discharge occurs due to a significant difference in electrical potential energy, creating a path of least resistance that carries charge.

When we see a flash of lightning, it's the rapid movement of electrons moving through this path, releasing energy in the form of light and heat. The passage of this colossal current can be modeled as a straight wire for calculation purposes. This model simplifies determining its magnetic and electric contributions to the surrounding environment.

To calculate the impact of a lightning bolt, often it is represented as a long, straight line of current - much like a conductor. This allows the application of fundamental physical laws, like the Biot-Savart Law, to estimate the magnetic fields generated by these mighty currents.

charge flow

Charge flow is simply the movement of electric charge from one point to another. This movement is the essence of what we call electric current. During a lightning strike, charge flows almost instantaneously due to the high voltages involved.

In terms of charge flow, the speed and quantity of electrons moving through the air create a significant amount of electrical energy. This flow can be immense, as evidenced by the calculation that a lightning bolt can carry a charge of \(15\) coulombs in a fraction of a second.

Understanding charge flow is crucial because it helps to explain how electric circuits function, and in nature, how phenomena like lightning develop. The enormity of the charge and the rapid rate of flow during a lightning event underscore the immense forces at play and why safety measures are critical during thunderstorms.