Answer

Verified

411k+ views

**Hint:**Initially the body of mass $M$ is oscillating about $x = 0$, this oscillation is simple harmonic oscillation because the body is attached to spring. Now, an electric field is applied in the x-direction.

This electric field will apply force on the body as it is charged. Consider the force on the body, find the acceleration of the body and look at changes occurring as an electric field is applied, accordingly determine the new equilibrium position and find the energy of the system.

**Complete step by step solution:**

The electric field applied is constant.

Consider the body to be moving in the positive x-direction and let the distance of the body from the mean position $x = 0$ be $x$.

The forces on the body will be due to the spring towards the mean position (negative direction) and due to the electric field in positive x-direction.

$F = - kx + qE$, because the force on a charged particle due to an electric field is given as charge multiplied by the electric field.

This force has to be equal to mass times acceleration according to laws of motion, we have,

$

Ma = - kx + qE \\

a = - \dfrac{k}{M}x + \dfrac{{qE}}{M} \\

\therefore a = - \dfrac{k}{M}\left( {x - \dfrac{{qE}}{k}} \right) \\

$

As you can see that this equation resembles the equation of SHM ($a = - \left( {\dfrac{k}{M}}

\right)x$). Hence, it is still an SHM but with a different mean position. The mean position is where the acceleration of the body becomes zero. So, we have $a = - \dfrac{k}{M}\left( {x -

\dfrac{{qE}}{k}} \right) = 0 \to x = \dfrac{{qE}}{k}$. Therefore, the new mean/equilibrium position is at a distance of $x = \dfrac{{qE}}{k}$ from $x = 0$.

Now, let us find the total energy of the system.

Consider the particle at its equilibrium position, the acceleration will be zero, the velocity will be maximum and is equal to $v = \omega A$ where $\omega $ is the angular frequency and \[A\] is equal to the amplitude. Also, we have a stretch of $\dfrac{{qE}}{k}$ in the spring (since, the position where the stretch is zero is $x = 0$). Therefore, the total energy is given by $T.E = K.E + {E_{spring}}$

$

T.E = \dfrac{1}{2}m{v^2} + \dfrac{1}{2}k{x^2} \\

T.E = \dfrac{1}{2}m{\left( {\omega A} \right)^2} + \dfrac{1}{2}k{\left( {\dfrac{{qE}}{k}} \right)^2}

\\

T.E = \dfrac{1}{2}m{\omega ^2}{A^2} + \dfrac{1}{2}\dfrac{{{q^2}{E^2}}}{k} \\

$

Here, for our ease, we have considered the total energy at the new equilibrium position. You can consider any position you want, you will get the same answer.

Therefore, the total energy of the system is $\dfrac{1}{2}m{\omega ^2}{A^2} +

\dfrac{1}{2}\dfrac{{{q^2}{E^2}}}{k}$ and the new equilibrium position is at a distance

$\dfrac{{qE}}{k}$ from $x = 0$.

**Option (A) is correct.**

**Note:**Whenever a constant electric field is applied, it only changes the mean position of the oscillating charged particle. It does not affect anything else. Remember how we used the force on the body to determine the new equilibrium position. Also keep in mind the conditions at mean position, like the values of acceleration and velocity. The total energy of the system is given as the sum of kinetic energy and the potential energy. In this case, the potential energy is the energy stored in the spring.

Recently Updated Pages

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Advantages and disadvantages of science

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

What organs are located on the left side of your body class 11 biology CBSE