Partnership for Integration of Computation into Undergraduate Physics

### Exercise Set 1 1. What is the total net force on the dipole? 2. What is the magnitude of the net torque on the dipole in terms of $q$, $d$ and $|\mathbf{E}|$? 3. Express the torque as a vector in terms of the vectors $\mathbf{p}$ and $\mathbf{E}$. 4. For what angles $\theta$ is the dipole * in stable equilibrium? + in unstable equilibrium? + undergoing maximum torque? ### Exercise Set 2 1. Using the small angle approximation (i.e. for $|\theta| \ll 1$ then $\sin(\theta) \approx \theta$), show that for small initial displacement, $\theta$, the electric dipole undergoes simple harmonic motion. (Hint: use your result from Exercise Set 1 and remember that $\tau = I\alpha$ where $\alpha$ is the angular acceleration and $I$ is the moment of inertia.) + What is the angular frequency of the oscillations? + Write out the formula for $\theta$ as a function of $t$. (Hint: this should be a solution to the second order differential equation above) + Edit the code below to plot a graph of the angle of displacement from equilibrium vs. time where $q=1.6\times 10^{-19}$ C, $d=5$ cm, $m = 1.67\times 10^{-27}$ kg, and $E=100$ N/C. ```python theta0=(10.0/180)*np.pi #initial displacement in radians q = #Charge on dipole in Coulombs d = #distance separating the charges (m) m = #mass of a single charge (kg) E = #magnitude of the electric field ############################################################ ######## EDIT BELOW WITH YOUR VALUE ########### ######## THIS IS THE ANGULAR FREQUENCY ########### omega = #enter your angular frequency ############################################################ ############################################################ ######## Determine the Period of the motion ######## ######## in terms of the angular frequency ######## T = #edit this for the period of motion # ############################################################ t=np.linspace(0,2*T,2000) #time interval from t = 0 to t = 2T theta = theta0*np.cos(omega*t) #solution for Simple harmonic motion ############################################################ ######## The following code plots the data ####### plt.plot(t,theta) plt.title("Simple Harmonic Motion - dipole in an Electric field") plt.xlabel("time (s)") plt.ylabel("angle (radians)") plt.grid() plt.show() ############################################################ ``` ###Exercise Set 3### 1. in the code below you can see some parameters which you can edit. Also, there is a field in which you need to put in your value of the angluar frequency. $\omega$ you obtained above. (a) Determine the time, for a dipole with $q=e$, $d=3\times 10^{-9}$ m, $m=3\times 10^{-5}$ kg in an electric field, $E=1000$ N/C, at which the small-angle solution differs from the non small-angle solution by 0.01 rads. 2. Try other parameter values as well. ```python ############################################################ ################ Start of parameters ################### theta_degrees = 30.0 #intial angle of displacement (degres) Tmax = 500.0 #upper bound for time (s) q = 1.0e-19 #Charge on dipole in Coulombs d = 3.0e-9 #distance separating the charges (m) m = 3.0e-5 #mass of a single charge (kg) E = 1000 #magnitude of the electric field N = 50000 #number of time intervals error = 0.05 ############################################################ ######## EDIT BELOW WITH YOUR VALUE ########### ######## THIS IS THE ANGULAR FREQUENCY ########### omega = np.sqrt((2.0*q*E/(d*m))) ## EDIT THIS ## ############################################################ ################ End of parameters ################### ############################################################ ############################################################ ############################################################ #Set the times to evaluate t = np.linspace(0,Tmax,N) #convert initial angle to radians theta_rads = theta_degrees * np.pi / 180.0 ############################## # compute approximate theta values on the time intervals Approx_Solution = theta_rads*np.cos(omega*np.array(t)) ############################## # Set the intial condition [initial angle, initial velocity] x0 = [theta_rads, 0.0] ############################## # numerically solve the real solution using odeint sol2 = odeint(torque,x0,t,args=(q,d,m,E)) ############################# # Plot the two solutions plt.plot(t,theta_rads*np.cos(omega*np.array(t))) plt.plot(t,sol2[:,0]) plt.grid() plt.xlabel("time (s)") plt.ylabel("angle (rads)") plt.show() ############################ # compute the difference between approx and actual on intervals diff = [np.abs((a1-b1)) for a1,b1 in zip(Approx_Solution,sol2[:,0])] ######################################### # Plot the difference vs time plt.plot(t,diff) plt.xlabel("time (s)") plt.ylabel("Difference") plt.show() ######################################### # Find when the two differ by .01 radians flag = True for i in range(len(t)-1): if abs(diff[i]> error): print("after t =", t[i],"the two solutions differ by more than .01 rads") flag = False break if flag: print("solution always within",error," rads") ``` 2. Try varying the magnitude of the electric field (e.g. $1000 \leq E \leq 5000$). What do you notice about the time it takes for the solutions to differ by .01 rads? 3. If you run the last cell, you will obtain a short animation displaying the dipole oscillating about the stable equilibrium point and below that a graph of the difference between the two different solutions and the approximation. What can you notice about when the differences are largest? Can you explain?