Role of Differential Equations in Physics

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Article By: Dr. Anees Rasheed Khan, Amravati
Jr. College Lecturer (Physics)

It is a well known fact that Mathematics is the language of Physics. We cannot understand Physics without the understanding of Mathematics. The knowledge of some basic mathematical terms like Limits, Derivatives, Integration, Differential Equations, ....etc is essential for understanding Physics properly. In this article, I am going to deal with the use of Differential Equations in Physics. For simplicity, we are dealing with only two totally different areas of Physics in which the concept of Differential Equations plays an important role.

A) Mechanical System:

 

 

Let a block of mass M is attached with one end of a spring as shown in the above figure. When the block is pulled from its equilibrium position through a small displacement ‘x’, then some restoring force ‘F’ will act on it. This force is given by,

                     F=M×a --------(1)

                    Now acceleration is given by,

                    ∵a= dv/dt

                   ∴F=M×(dv/dt)

                   Now velocity is given by,

                   ∵v= dx/dt

                   ∴a= (d/dt)×(dx/dt)=(d^2 x)/dt^2

                   ∴F=M×(d^2 x)/dt^2 -------(2)

                Now, let the spring constant is ‘K’ then the force of stretching produced in the spring is given by,

                  F∝=-x

                 F=-K.x

                 Putting this value in equation (2) we get,

                 M×(d^2 x)/dt^2 =-K.x

                 M×(d^2 x/dt^2)+K.x=0

                (d^2 x/dt^2) +(K/M).x=0

This is the differential equation for the oscillatory motion of a block attached with a spring. By solving this differential equation we can get the solution for the value of displacement ‘x’ as follows,

                 x=A sin(√(K/M) t)

Here A is amplitude (maximum value of displacement) of oscillations of the block on the frictionless surface.

B) Electrical System:

 

Let a capacitor ‘C’ and an inductor ‘L’ are connected with each other in parallel combination to form a tank circuit as shown in the above figure. In this tank circuit, when the switch is closed, then the capacitor will be discharged and current flows through the coil L. Due to change in current passing through the coil, an emf is generated there, which causes the capacitor to charge again. In this way, the electrical oscillations are produced in the LC tank circuit.

               The voltage across the capacitor is given by,

               V_c=-q/C       ------from the definition of capacity

               The voltage (induced emf) across the inductor is given by,

               V_l=-L.dI/dt       ------by using the concept of self inductance

              The overall potential in the tank circuit is conserved. It means that the sum of all potentials in the circuit will be zero.

              ∑V=0

              Vc+Vl=0

              (-q/C)-L.(dI/dt)=0

              (q/C)+L.(dI/dt)=0-------------(1)

             Now, using the concept of electric current, we have

             I=dq/dt

             Putting this value in equation (1), we get

             q/C+L.(d/dt)(dq/dt)=0

             q/C+L.(d^2 q)/(dt)^2 =0

             L.(d^2 q)/(dt)^2 +q/C=0

            (d^2 q)/(dt)^2 +(1/LC)q=0

            This is the differential equation for the electrical oscillations of LC tank circuit. By solving this differential equation we can get the solution for the value of charge ‘q’ as follows,

             q=A sin(1/√LC t)

 Here A is amplitude (maximum value of charge) of electrical oscillations of LC tank circuit.
     From the above two totally different areas of Physics, we have studied the use of Differential Equations in Physics. Hence the study of Differential Equations and their solutions are of great importance in Physics.