Question

The magnetic potential \(P\) at a point on the axis of a circular coil is given by \(P=\frac{2 \pi N I r}{k} \int_{c}^{\infty} \frac{1}{\left(r^{2}+x^{2}\right)^{3 / 2}} d x\) where \(N, I, r, k,\) and \(c\) are constants. Find \(P\)

Short answer

The magnetic potential \(P\) is given by \( P = \frac{2 \pi N I }{k r^2}[\pi/2 - tan^{-1}(c/r)].\)

Step-by-step solution

Recognize the Kind of Integral

The given integral is present in the form of \( \int dx/(a^2 + x^2)\). So, it is possible to evaluate it using the standard result \( \int dx/(a^2 + x^2) = (1/a) tan^{-1}(x/a) + C\) where in this case \(a = r\).

Evaluate the Integral

Substitute \(a = r\) in the above standard result and evaluate the integral, applying the limits of \(c\) and \(\infty\). Thus, \( \int_{c}^{\infty} \frac{1}{\left(r^{2}+x^{2}\right)^{3 / 2}} dx = \int_{c}^{\infty} \frac{1}{r^3} \cdot \frac{1}{\left(1+(x/r)^{2}\right)^{3 / 2}} dx = \frac{1}{r^3}\) [\(tan^{-1}(x/r)\) from \(c\) to \(\infty\)].

Apply the Limits

Now, we apply the limits \(\infty\) and \(c\). So, \( \frac{1}{r^3}[tan^{-1}(\infty) - tan^{-1}(c/r)] = \frac{1}{r^3}[\pi/2 - tan^{-1}(c/r)].\)

Substitute the Result back into the Given Expression

The magnetic potential \(P\) then becomes \( P=\frac{2 \pi N I r}{k} \cdot \left[\frac{1}{r^3}[\pi/2 - tan^{-1}(c/r)]\right]\). After simplifying, we get \( P = \frac{2 \pi N I }{k r^2}[\pi/2 - tan^{-1}(c/r)],\)

Key concepts

Integral Calculus

Integral calculus is a branch of mathematics focused on the accumulation of quantities and the areas under and between curves. When we discuss the integration of functions, we are essentially looking for the anti-derivative or inverse operation of differentiation. This becomes crucial in physics and engineering applications where the goal is to calculate quantities like distance, area, volume, and—in our specific problem—the magnetic potential.

When facing an integral, we normally start by identifying the form of the integral and check if it corresponds to any known standard results. These standard results act as shortcuts that allow us to solve the integrals without actually performing the operation of integration each time. In our exercise, the problem involved a definite integral which is an integral with specific limits. Recognizing that it matches the form of the standard result for the arctangent function greatly simplifies our work.

Magnetic Potential of a Circular Coil

The concept of magnetic potential is analogous to electric potential, representing a scalar quantity which, when we take the gradient, gives us the magnetic field. Specifically, for a circular coil, the magnetic potential at a point depends on the coil's geometry, the number of turns, the current passing through it, as well as the permissivity of the medium around it. The potential at a point along the axis of a coil is related to the coil’s radius and the distance from the coil to the point where we're measuring.

To visualize this, imagine the circular coil as a clock face and the point where we measure potential is a hovering fly—the potential tells us how strong the magnetic 'wind' is at the point where the fly is. Our exercise addresses this by setting up an integral that models the effect of every part of the coil on the fly, summing it up to find the total potential.

Definite Integrals

Definite integrals extend the concept of area under a curve to provide net 'signed' areas, which means areas above the x-axis are positive and those below are negative. The power of definite integrals is in their ability to quantify total change, a core aspect in physics and engineering problems.

In our exercise, the definite integral represents the accumulated effect—from a certain starting point to infinity—of the coil on the magnetic potential at a point. Evaluating a definite integral involves finding the indefinite integral (also known as the antiderivative) followed by applying the Fundamental Theorem of Calculus, which allows us to subtract the antiderivative evaluated at the lower limit from the antiderivative evaluated at the upper limit. This step, often referred to as 'applying the limits', is crucial as it provides us with the exact value of the magnetic potential, encapsulating the concept of 'from c to infinity' in our problem.