Question

What is a good rule of thumb for designing a simple magnetic coil? Specifically, given a circular coil of radius \(\sim 1 \mathrm{~cm},\) what is the approximate magnitude of the magnetic field, in gausses per amp per turn? a) \(0.0001 \mathrm{G} /(\mathrm{A}-\mathrm{turn})\) c) \(1 \mathrm{G} /(\mathrm{A}\) -turn \()\) b) \(0.01 \mathrm{G} /(\mathrm{A}-\mathrm{turn})\) d) \(100 \mathrm{G} /(\mathrm{A}-\mathrm{turn})\)

Short answer

Answer: The rule of thumb for designing a simple magnetic coil is to use the Biot-Savart Law to calculate the magnetic field. For a circular coil of radius 1 cm, the approximate magnitude of the magnetic field is 1 G/(A-turn).

Step-by-step solution

Recall Biot-Savart Law

The Biot-Savart Law states that the magnetic field \(\vec{B}\) produced by a small current element \(\vec{I}\) at a point \(\vec{r}\) in space is given by: $$\mathrm{d}\vec{B} = \frac{\mu_0}{4 \pi} \frac{ I (\mathrm{d}\vec{l} \times \vec{r})}{r^3}$$ where \(\mu_0\) is the permeability constant, \(\mathrm{d}\vec{l}\) is the incremental length vector, and \(r\) is the distance from the current element to the point in space.

Calculate the Magnetic Field for a Single Loop

For a single circular loop, the magnetic field at the center can be calculated by integrating the expression from Biot-Savart Law over the entire loop. $$\vec{B} = \frac{\mu_0 I}{4 \pi} \int \frac{(\mathrm{d}\vec{l} \times \vec{r})}{r^3}$$ Using symmetry, the expression becomes: $$B = \frac{\mu_0 I}{2 R}$$ where R is the radius of the circular loop.

Convert to Gauss Per Ampere Per Turn

The magnetic field \(B\) has units of Tesla. To convert it to Gauss, we can use the following conversion: $$1 \mathrm{T} = 10^4 \mathrm{G}$$ For one turn and one ampere, the magnetic field is given by: $$B_{1A,1t} = \frac{\mu_0}{2 R} \times 10^4 \mathrm{G}$$

Substitute the numerical values

The permeability constant \(\mu_0 = 4 \pi \times 10^{-7} \mathrm{T.m/A}\) and the radius of the circular loop is given as \(R = 1 \mathrm{cm} = 0.01 \mathrm{m}\) Substituting these values, we get the expression for the magnetic field: $$B_{1A,1t} = \frac{4 \pi \times 10^{-7} \times 10^4}{2 \times 0.01} \mathrm{G/(A-turn)}$$

Simplify the expression

$$B_{1A,1t} = \frac{4 \pi}{2 \times 0.01} \mathrm{G/(A-turn)} = \frac{2 \pi}{0.01} \mathrm{G/(A-turn)} \approx 628.32 \times 10^{-3} \mathrm{G/(A-turn)} \approx 0.63 \mathrm{G/(A-turn)}$$ Based on this result, the closest approximate magnitude of the magnetic field for a radius \(~1\mathrm{cm}\) is: c) \(1 \mathrm{G} /(\mathrm{A}\) -turn \()\)

Key concepts

Biot-Savart Law

Understanding the Biot-Savart Law is essential for calculating the magnetic field created by current-carrying conductors. This fundamental law in electromagnetism relates the magnetic field \( \vec{B} \) to the magnitude, direction, length, and proximity of an electric current.

The law is mathematically expressed as \[ \mathrm{d}\vec{B} = \frac{\mu_0}{4 \pi} \frac{ I (\mathrm{d}\vec{l} \times \vec{r})}{r^3} \], where:\

• \( \mu_0 \) is the permeability of free space,
• \( I \) is the electric current,
• \( \mathrm{d}\vec{l} \) is an infinitesimally small segment of the conductor over which the current is flowing,
• And \( \vec{r} \) is the vector from the element \( \mathrm{d}\vec{l} \) to the point where the magnetic field is being calculated.

To find the total magnetic field, this law requires integrating over the length of the conductor. For a simple circular coil, as in a magnetic coil exercise, the symmetry simplifies the integration, resulting in a more straightforward formula to work with. This simplification leads to a clear understanding that the magnetic field directly in the center of a circular loop depends inversely on the radius of the loop — the smaller the radius, the stronger the magnetic field at its center.

Permeability Constant

The permeability constant \( \mu_0 \) is a physical constant that is central to the calculation of the magnetic field in vacuum due to an electric current. It is defined as the measure of the ability of a vacuum to support the formation of a magnetic field within itself.

In the International System of Units (SI), the permeability constant is given as \( \mu_0 = 4 \pi \times 10^{-7} \mathrm{T.m/A} \) (Tesla meter per Ampere). This constant bridges electric currents to the resulting magnetic fields and is used in conjunction with the Biot-Savart Law to calculate the magnetic influence of the current.

When applying \( \mu_0 \) to our magnetic coil calculation, it acts as a scaling factor, determining how a given current and coil size will translate into a magnetic field strength. By substituting values into \( \mu_0 \) and the coil's radius, we can derive the intensity of the magnetic field for a specific setup.

Gauss per Ampere per Turn

Gauss per Ampere per Turn is a useful unit of measure when designing and understanding magnetic coils. It provides a direct relationship between the magnetic field strength (\( B \) in Gauss), the electric current (\( I \) in Amperes), and the number of turns in the coil.

To convert from Tesla, which is the SI unit for magnetic field strength, to Gauss, we use the relation \[ 1 \mathrm{T} = 10^4 \mathrm{G} \. With this conversion, magnetic field calculations performed using the SI unit can be easily translated to Gauss per Ampere per Turn.

In the context of our example with a single circular coil, knowing the value in Gauss per Ampere per Turn is particularly valuable for practical applications such as designing electromagnets or inductive sensors, where it is critical to predict how much magnetic field will be generated for a given current in a coil of a certain number of turns. By calculating the theoretical value and comparing it to the given options, we determined that the correct answer is approximately 1 Gauss per Ampere per Turn, which is directly applicable in these practical scenarios.