8. Circuits and Magnetism I

The professor begins by revisiting circuits, emphasizing the complexity beneath the surface of a simple battery and resistor setup. They explain that while electrostatic force is involved, it's not the sole player in the circuit's operation. The work done in the circuit, such as heating a resistor, suggests another force at work within the battery. An analogy is drawn between a ski lift and a circuit, illustrating how the electromotive force (EMF) is akin to the mechanical work done against gravity by the ski lift. The concept of EMF is introduced as the line integral of forces around a loop, which, unlike electrostatic force, is non-zero due to non-conservative forces within the battery.

The discussion continues with the exploration of simple circuits involving a single resistor and an electromotive force (EMF) source. The professor outlines the fundamental principles that govern these circuits, such as the conservation of current and the relationship between voltage, current, and resistance (Ohm's Law). They then extend the analysis to more complex circuits with multiple resistors in series and parallel, explaining how to calculate the equivalent resistance and the current flow through each resistor. The concept of emf as the voltage difference around a closed loop is also clarified.

The professor moves on to capacitors, explaining how they behave when placed in series and parallel. They describe the calculation of equivalent capacitance in both configurations, highlighting the difference in behavior compared to resistors. The analogy of a ski slope is used again to illustrate the concept of potential difference and energy storage in capacitors. The lecture also touches on the idea of energy conservation in a circuit involving a battery, resistor, and capacitor, setting the stage for a homework problem.

The focus shifts to the behavior of capacitors when they are charged in a circuit. The professor describes the process of charging a capacitor with a resistor and a battery, detailing the current flow and the resulting charge on the capacitor over time. A differential equation is derived to describe the charge as a function of time, leading to an exponential growth model for the charge on the capacitor. The final charge on the capacitor is shown to be dependent on the capacitance and the EMF of the battery. The lecture concludes with a student's question about the negative sign in the equation, which the professor clarifies.

The professor delves into the energetics of the circuit involving a capacitor, a resistor, and a battery. They ask students to consider the initial and final states of the system to understand the energy transfer. The work done by the battery is shown to be equal to the energy stored in the capacitor and the energy dissipated in the resistor. The current's exponential decay with time is derived from the charge function, and the homework assignment is outlined, which involves calculating the energy given by the battery and ensuring it equals the sum of the energy stored and dissipated.

The lecture concludes with a discussion on analyzing more complex circuits with multiple branches and resistors. The professor emphasizes the importance of identifying the number of unknowns and setting up the correct number of independent equations to solve the circuit. The technique of using loop equations based on voltage drops and Ohm's law is introduced. The professor also touches on the concept of reference frames and their relation to the observed phenomena in the circuit, hinting at the principles of relativity.

The professor introduces magnetism as the next topic, starting with historical discoveries and moving on to describe new phenomena not covered by electrostatics. They discuss the forces observed around moving charges and magnets, such as the attraction or repulsion between a moving charge and a neutral wire. The concept of reference frames in relation to magnetism is also mentioned, with a promise to delve deeper into relativistic effects in future lectures.

The professor explores the concept of magnetism from the perspective of a moving observer. They discuss the scenario where a charge in a moving train is attracted to a wire and use it to illustrate the principles of relativity. The key takeaway is that electromagnetism works for all observers in uniform motion, and the laws of physics remain the same. The lecture ends with a teaser for the fundamental equations of magnetostatics that will be covered in subsequent classes.

The professor presents the fundamental equations of magnetostatics, which describe the force on a moving charge in a magnetic field and how electric currents produce magnetic fields. The Lorentz force, given by the cross product of velocity and magnetic field, is introduced as a postulate based on experimental evidence. The professor also discusses the concept of work done by the magnetic field, highlighting that it always perpendicular to the motion of the particle and thus does no work. The lecture includes simple problems to illustrate the application of the v x B force.

The lecture focuses on applications of magnetic fields in particle physics, specifically velocity filtering and particle trapping. The professor explains how a combination of electric and magnetic fields can be used to select particles with a specific velocity, creating a velocity filter. They also describe how a uniform magnetic field can be used to trap particles in a circular orbit, with the frequency of the orbit dependent only on the charge-to-mass ratio of the particle and the magnetic field's strength. The historical use of this principle in the invention of the cyclotron by Lawrence is also mentioned.

The professor discusses the force experienced by a small segment of a wire carrying current in a magnetic field. They derive the force as the product of current, the length of the wire segment, and the magnetic field, taking the cross product into account. The lecture explores scenarios where this force can be used to balance the weight of the wire or to distort a loop of current-carrying wire. The concept of torque on a current loop in a magnetic field is introduced, with the magnetic moment of the loop being analogous to a magnetic dipole moment.

The final topic of the lecture is the calculation of torque on a rectangular current loop in a magnetic field. The professor explains how the forces on different segments of the loop result in a net torque, which is equivalent to the magnetic moment of the loop (current times the area) cross the magnetic field. The magnetic moment is a vector quantity, and its interaction with the magnetic field is similar to that of a magnetic dipole in a field. The lecture concludes with a setup for the next class, where further exploration of this topic is expected.