[2018EM/HMW-06]

Electrodynamics                                                                         Oct 24, 2018

Tutorial-VI

• Show that the sign of time component of a time like vector remains the same in all Lorentz frames. Is this statement true for space like frames?
• If \(p^\mu=(p^0,0,0,p)\) with \( p^{02} -\vec{p}^2 = m^2\), find all vectors \(\epsilon^\mu\) such that \(\epsilon^\mu p_\mu=0\). How many linearly independent vectors can you make?
• If \(A_{\lambda\sigma}\) and \(B_{\lambda\sigma}\) are two second rank tensors satisfying \[ A_{\lambda\sigma}x^\lambda x^\sigma = B_{\lambda\sigma}x^\lambda x^\sigma\] for arbitrary four vector \(x^\mu\), what is the relationship between \(A_{\lambda\sigma}\) and \( B^{\lambda\sigma}\)?
• Let \(\epsilon\) and \(\lambda\) be two light like four vectors. What is the range of allowed values of \(\epsilon\cdot\lambda\)? Express your answer in terms of \(\epsilon^0\) and \(\lambda^0\).
• A uniform magnetic field \(\vec{B}\), pointing straight up, fills the shaded circular region of Fig.7.24. If \(\vec{B}\) is changing with time, what is the induced electric field?
• A line charge \(\lambda\) is glued onto the rim of a wheel of radius \(b\), which is then suspended horizontal as shown in Fig. 7.25, so that it is free to rotate (the spokes are made of some nonconducting material—wood, maybe). In the central region, out to radius \(a\), there is a uniform magnetic field \(B_0\), pointing up. Find the angular momentum of the disc when the magnetic field is switched off. Induction