spherical shell

Spherical shell potential (Griffiths)


The charged sphere A spherical shell has spherical symmetry and because of this we write its potential as $V(R,\theta)=V_0(\theta)$ ($\theta$ is the “polar” angle of the spherical coordinate system). a) When $V_0(\theta)=V_0(1-\theta^2)$ find the equation of definition for $V(R,\theta)$ if $r>R$. b) When $V_0(\theta)=V_0(\cos^2 \theta +2)$ please evaluate the potential of this spherical shell at

simple tunneling

Simple Tunneling


Electron wave An electron travels toward negative values of $x$ having a kinetic energy $E=2 eV$. Please write the equation of this electron wave and describe its characteristic values. The general equation of the wave travelling in the negative x direction (towards left) is $\psi(x,t)=A\exp[i(kx+\omega t)]$  ($A$ is the wave amplitude) For a fixed point

Infinite Slot Potential


What is the potential of the infinite slot from Griffiths 3.3, if you are given a varying potential $V(0,y)=V_0x/a$ as a boundary condition at $x=0$ For the given arrangement of example 3.3, the general solution of the potential that satisfies the 3 zero boundary conditions ($V=0$ in $y=0$, in $y=a$ and when $\to \infty$) is$V(x,y)=\sum_{n=1}^{\infty}C_n

Monopoles and Dipoles


You are given two charges $q_0$ and $q$ situated on the z axis at $z=+a$ and $z=-a$. What are the monopole and dipole terms in the potential, when the distance from origin $r>>a$ ifa) $q=q_0$b) $q=-q_0$c) $q=0$$d=2a$$R_±=r^2+(d/2)^2∓d*r*\cos⁡ θ= r^2 [1∓d/r*\cos⁡θ+d^2/(4r^2 )]$$1/R_± =(1/r)*\frac{1}{√((1∓d/r*cos⁡〖θ+d^2/(4r^2 ))}≈(1/r)*(1/\sqrt{1∓d/r*cos⁡θ})≈$$≈(1/r)*[1∓(-1/2)*d/r  \cos⁡θ]$         (1) If $q=q0$:  $V(r)=k(q/R_+ +q/R_- )$$1/R_+ +1/R_- =1/r*[1+⋯.]+1/r*[1-….]=2/r$$V(r)=2kq/r$So that this

Rotating Square Loop


For a square loop of side $a$ in a magnetic field $\mathbf{B}=B_0[1-(x^2/l^2)]*\hat x$ rotating with angular velocity $\omega$ please find $E(t)$ induced in loop. In text it says that the plate is square so I guess l=a in the definition of B$B=B_0 (1-x^2/l^2 )*x ̂$Cylindrical coordinates$x=r*\cos(ωt)$  and $y=r*\sin(⁡ωt)$$dS=a*dr$The flux is$dϕ=B ⃗*dS ⃗=B_0 (1-(r^2  \cos^2⁡ωt)/l^2 )*(a*dr)*\cos

Step Down Potential (Griffiths)


Find the reflection coefficient of a plane wave with energy $E>V$ on a “step potential” initially 0 that drops down to $-V_0$. How is this similar and different to a classical particle?  Consider a plane wave coming from the left towards the step. Since $E>0$ it is a wave that has two components: incoming component

Cylinder with Uniform Current


For a uniform current that goes through a long cylinder (z-axis) please find a) force exerted by the upper half on the lower half of the cylinder. b) magnitude of this force for cylinder length of 1 meter, diameter of 1 mm and current of 10 A.  The current density inside the wire is$j=I/S=I/(πa^2)$Consider first the