Physical Applications of Line Integrals

In physics, the line integrals are used, in particular, for computations of

mass of a wire;
center of mass and moments of inertia of a wire;
work done by a force on an object moving in a vector field;
magnetic field around a conductor (Ampere's Law);
voltage generated in a loop (Faraday's Law of magnetic induction).

Consider these applications in more details.

Mass of a Wire

Suppose that a piece of a wire is described by a curve $C$ in three dimensions. The mass per unit length of the wire is a continuous function $ρ (x, y, z) .$ Then the total mass of the wire is expressed through the line integral of scalar function as

m = \int_{C} ρ (x, y, z) d s .

If $C$ is a curve parameterized by the vector function $r (t) =$ $(x (t), y (t), z (t)),$ then the mass can be computed by the formula

m = \int_{α}^{β} ρ (x (t), y (t), z (t)) \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2} + {(\frac{d z}{d t})}^{2}} d t .

If $C$ is a curve in the $x y$ -plane, then the mass of the wire is given by

m = \int_{C} ρ (x, y) d s,

or in parametric form

m = \int_{α}^{β} ρ (x (t), y (t)) \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}} d t .

Center of Mass and Moments of Inertia of a Wire

Let a wire is described by a curve $C$ with a continuous density function $ρ (x, y, z) .$ The coordinates of the center of mass of the wire are defined as

\bar{x} = \frac{M_{y z}}{m}, \bar{y} = \frac{M_{x z}}{m}, \bar{z} = \frac{M_{x y}}{m},

where

M_{y z} = \int_{C} x ρ (x, y, z) d s, M_{x z} = \int_{C} y ρ (x, y, z) d s, M_{x y} = \int_{C} z ρ (x, y, z) d s

are so-called first moments.

The moments of inertia about the $x$ -axis, $y$ -axis and $z$ -axis are given by the formulas

Work

Work done by a force on an object moving along a curve is given by the line integral

where is the vector force field acting on the object, is the unit tangent vector (Figure ).

Work done by a force F on an object moving along a curve C — Figure 1.

The notation means dot product of and

Note that the force field is not necessarily the cause of moving the object. It might be some other force acting to overcome the force field that is actually moving the object. In this case the work of the force could result in a negative value.

If a vector field is defined in the coordinate form

then the work done by the force is calculated by the formula

If the object is moved along a curve in the -plane, then the following formula is valid:

where

If a path is specified by a parameter ( often means time), the formula for calculating work becomes

where goes from to

If a vector field is conservative, then then the work on an object moving from to can be found by the formula

where is a scalar potential of the field.

Ampere's Law

The line integral of a magnetic field around a closed path is equal to the total current flowing through the area bounded by the contour (Figure ).

This is expressed by the formula

where is the vacuum permeability constant, equal to

Faraday's Law

The electromotive force induced around a closed loop is equal to the rate of the change of magnetic flux passing through the loop (Figure ).

Solved Problems

Example 1.

Find the mass of a wire running along the plane curve with the density The curve is the line segment from point to point

Solution.

We first find the parametric equation of the line

where parameter varies in the interval Then the mass of the wire is

Example 2.

Find the mass of a wire lying along the arc of the circle from to with the density (Figure ).

Solution.

A wire lying along the arc of the circle x^2+y^2=10 — Figure 4.

The circle with radius and centered at the origin has parametric equations

where the parameter varies in the interval Then the mass of the wire is calculated as follows:

See more problems on Page 2.