5.4. Applicatins in Physics#
Integrals appear everywhere in physics but many of them are too difficult to calculate analytically. Here are some examples.
Electric potential generated by a charge on a ring (Section 5.4.1): Electric charge distributed on a ring generates an electric potential. By solving the Laplace equation, we can find the potential analytically up to an integral. The mathematical analysis is stuck and a numerical method is needed to complete the final step.
The period of classical oscillation (Section 5.4.3): A classical particle bound in a potential oscillates between two turning points as discussed in Section 4.4.1. The period of oscillation can be expressed as an integral but the integrand has two integrable singularities at the turning points. The methods discussed in Section 5.3.2 must be used.
Scattering Angle (Section 5.4.4): When a particle collides with another particle, its trajectory is deflected to a new direction. The angle of the deflection can be expressed by an integral, which is doubly improper. The integration interval is unbound and the integrand has a integrable singularity. Therefore, this is a challenging integral for numerical calculation.
Thermodynamics of quamtum ideal gases (Section 5.4.2): Some thermodynamic relations for Fermi gases and Bose gases are expressed with integrals known as the Fermi-Dirac integral and the Bose-Einstein integral, respectively. The integrals are improper since the upper bound in the integrals is infinity. WE use the methods discussed in Section 5.3.1.
Updated on 4/16/2024 by R. Kawai