Update extra_exercises.md - added exercise that was removed from Drude lecture

docs/extra_exercises.md

@@ -88,4 +88,19 @@ $$E_{vb}(k_x, k_y)=2t_{vb} [\cos(k_x a)+\cos(k_ya)-2].$$
 
     We now add doping to the semiconductor by adding $n_D$ donor atoms per unit area and $n_A$ acceptor atoms per unit area to the semiconductor.
 
-5. Assume in this question that $t_{cb} = t_{vb} = t$. Derive an expression for $\mu$ of the semiconductor in the case of doping. You may assume that all the donor atoms are ionized and all acceptor atoms are occupied.
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+5. Assume in this question that $t_{cb} = t_{vb} = t$. Derive an expression for $\mu$ of the semiconductor in the case of doping. You may assume that all the donor atoms are ionized and all acceptor atoms are occupied.
+
+### Exercise 7: Motion of an electron in a magnetic and an electric field.
+Consider an electron in free space experiencing a magnetic field $\mathbf{B}$ along the $z$-direction.
+%Assume that the electron starts at the origin with a velocity $v_0$ along the $x$-direction.
+
+#### Question 1.
+Write down the Newton's equation of motion for the electron, compute $\frac{d\mathbf{v}}{{dt}}$.
+#### Question 2.
+What is the shape of the motion of the electron? Calculate the characteristic frequency and time-period $T_c$ of this motion for $B=1$ Tesla.
+#### Question 3.
+Now we accelerate the electron by adding an electric field $\mathbf{E} = E \hat{x}$. Adjust the differential equation for $\frac{d\mathbf{v}}{{dt}}$ found in (1) to include $\mathbf{E}$. Sketch the motion of the electron.
+#### Question 4.
+Consider now an ensemble of electrons in a metal. Include the Drude scattering time $τ$ into the differential equation for the velocity you formulated in 3.
+
+Note that the differential equation now describes the *average* velocity of the electrons in the ensemble.