Question

Suppose that the magnetic field of the Earth were due to a single current moving in a circle of radius \(2.00 \cdot 10^{3} \mathrm{~km}\) through the Earth's molten core. The strength of the Earth's magnetic field on the surface near a magnetic pole is about \(6.00 \cdot 10^{-5} \mathrm{~T}\). About how large a current would be required to produce such a field?

Short answer

Answer: The current required is approximately 1.91 × 10^7 A.

Step-by-step solution

Recall the formula for the magnetic field strength of a circular loop

The magnetic field strength (B) at the center of a circular current loop of radius (R) is given by the formula: B = \cfrac{\mu_0 I}{2R} Where I is the current in the loop, and \mu_0 is the permeability of free space, approximately 4π × 10^{-7} T m/A.

Rearrange the formula to solve for the current I

We can rearrange the formula for the magnetic field strength to solve for the current (I) as follows: I = \cfrac{2RB}{\mu_0}

Substitute the known values into the formula and calculate the current

We are given the radius (R) as 2.00 × 10^3 km = 2.00 × 10^6 m, and the magnetic field strength (B) as 6.00 × 10^{-5} T. We can substitute these values, along with the permeability of free space, into the formula and calculate the current (I): I = \cfrac{2(2.00\times10^{6}\mathrm{m})(6.00\times10^{-5}\mathrm{T})}{4\pi\times10^{-7}\mathrm{Tm/A}}

Perform the numerical calculation

Multiplying the values and simplifying the units, we can find the required current: I = \cfrac{2.4\times10^{1}\mathrm{TA}}{4\pi\times10^{-7}\mathrm{Tm/A}} = 2.4\times10^{1}\mathrm{TA} \times \cfrac{1}{4\pi\times10^{-7}\mathrm{Tm/A}} = 1.91 \times 10^{7} A

Write down the result

The current required to produce the given magnetic field strength near a magnetic pole is approximately 1.91 × 10^7 A.

Key concepts

Magnetic Field Strength

Understanding magnetic field strength is crucial when delving into electromagnetism. It's a measure of the magnetic force concentration at a given point in space, typically measured in Teslas (T) in the International System of Units (SI). Imagine the magnetic field as lines moving from the north pole to the south pole of a magnet; where the lines are closer together, the field strength is greater.

When considering Earth's magnetic field, the strength varies depending on the geographic location. Near the magnetic poles, it is stronger and can be approximated to be around 6.00 x 10^-5 T. This information is vital when assessing the required current to generate a similar field using electric currents, as done in the example problem. Just like a bar magnet with its field lines, the Earth generates its magnetic field, which is crucial for navigation, protecting us from solar wind, and a host of other ecological and technological processes.

Permeability of Free Space

The permeability of free space, denoted by the symbol \( \mu_0 \), is a fundamental physical constant. It describes how easily a magnetic field can permeate through the vacuum of space. Essentially, it is the degree of magnetization that a material obtains in response to an applied magnetic field.

In our calculations, \( \mu_0 \) plays a pivotal role as it links the electric current flowing through a wire (or loop) to the resulting magnetic field. Its value is approximately 4\pi x 10^-7 Tm/A. With this constant, we can calculate the magnetic effects of electric currents on a theoretical level and apply these calculations to real-world problems like determining the hypothetical electric current that could generate Earth's magnetic field.

Magnetic Field of Current Loop

A current loop generates a magnetic field with a pattern similar to that of a simple bar magnet and is described by its own set of rules. The strength of the magnetic field at the center of a circular current loop is a function of the current flowing through the loop, the radius of the loop, and the permeability of the free space.

The formula for this is \( B = \cfrac{\mu_0 I}{2R} \), where \(B\) is the magnetic field strength, \(I\) is the current, and \(R\) is the radius. In the context of the Earth-like magnetic field problem, we imagine the Earth's core as such a loop with current flowing through it, and we use this formula to back-calculate the necessary current. The simplicity of the formula belies the complex interaction of electric current and magnetic fields in space, reflecting one of the many elegant symmetries in physics.

Magnetic Field Calculation

Calculating the magnetic field involves using the aforementioned principles together with mathematical formulas specific to the scenario. For instance, with a simple cylindrical current loop, the Biot-Savart Law could be used. However, the provided exercise simplifies the process considerably, giving us a direct proportionality that delineates magnetic field strength in terms of the current and loop radius.

By rearranging the formula to solve for the current when the magnetic field strength and loop radius are known, you can evaluate the necessary realities to recreate certain field conditions. This process demonstrates the practical side of theoretical physics, allowing us to retrofit our formulas to imagine how nature works, or even to assess what's required to mimic natural phenomena, such as the Earth's magnetic field, with human-engineered solutions.