Charging a Capacitor in RC Circuits

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1 Lecture 8-18 Charging a Capacitor in RC Circuits 1. Charging equation: From Kirchhoff's Law q ε t/ RC t/ τ ε ir = 0 i = e = I0e C R 2. Switch closed at t = 0. Initially C is uncharged. ΔV C across C is initially zero. ΔV across R is ε. R ε thus i = at t = 0 R 3. During charging: As C is charged more and more ΔV C across C increases. At the same time ΔV R across R decrea ses. 4. Fully charged: Current is zero, thus Δ V = ε ΔVR thus C across R is zero, i = 0 at t = ε + _ ΔV C i t/ τ q C I Q C Discharging and charging e i ΔV R

2 Lecture 8-28 Charging Behavior of Capacitors Initially, the capacitor behaves like a wire because: ε i = at t = 0 R After a long time, the capacitor behaves like an open switch, because: i = 0 at t = Discharging Initially, the capacitor behaves like a battery, because: ε V Q i = = = at t = 0 R R RC After a long time, the capacitor behaves like an open switch, because: i = 0 at t =

Lecture 8-38 Magnetism In ancient Greece, iron ore was mined. Some were permanent magnets.

No single magnetic charge or stationary electric charge interaction with magnets or magnetic field has been

Interactions - among magnets, with N-S poles of the earth, with materials such as iron, nickel and with moving

3 Lecture 8-38 Magnetism In ancient Greece, iron ore was mined. Some were permanent magnets. This has been known for 2500 year. In China, compass needle was used to navigate at least a thousand years ago. No single magnetic charge or stationary electric charge interaction with magnets or magnetic field has been observed. Interactions - among magnets, with N-S poles of the earth, with materials such as iron, nickel and with moving charges have been observed. Magnetism is not a contact but an action at a distance interaction like E. Thus magnetic interaction is transmitted by magnetic field. The magnitude (strength) of B field is measured by interaction with a moving charge q.

4 Lecture 8-48 Magnetic Field B We will define magnetic field more precisely a few slides later. But for now, let us proceed to conceptualize it using the way magnets interact with other magnetic materials: B represents the way magnetic forces act at a distance from their sources B is a vector field, represented by field lines. Iron filling lined up with the magnetic filed of a bar magnet. The density of magnetic field lines represents the strength of B. The direction of magnetic field lines are tangent to the magnetic field and it is the direction other magnets like to align with.

5 Lecture 8-58 More Permanent Magnets and Magnetic Field Lines Bar magnet Earth Magnetic Field Magnetic Card C magnet Hose shoe magnet Can have many poles!

dipole. B F B on moving charges No sources or sinks Electric field lines of an electric dipole. Source of electric field lines.

6 Lecture 8-68 Magnetic Field Lines Bar magnet... two poles: N and S Like poles repel; Unlike poles attract. Magnetic Field lines: (defined similarly as electric field lines using the direction and density) Magnetic Field lines of a magnetic dipole. B F B on moving charges No sources or sinks Electric field lines of an electric dipole. Source of electric field lines. Sink of electric field lines. From North to South outside But opposite inside! Magnetic field lines form closed loops. No sinks or sources of field line. electric dipole

An entity which carried such magnetic charge would be called a magnetic monopole (having +

7 Lecture 8-78 Magnetic Monopoles Does there exist magnetic charge, just like electric charge? An entity which carried such magnetic charge would be called a magnetic monopole (having + or - magnetic charge). How can you isolate this magnetic charge? Try cutting a bar magnet in half. In fact no attempt has been successful in finding magnetic monopoles in nature.

Find: [ B] E = E + E = q E+ v B total E B F = qv B B This defines B.

8 Lecture 8-88 Magnetic Field B Defined When magnetic and electric forces act on a moving charge q: Vary q and v in the presence of a given magnetic field and measure magnetic force F on the charge. Find: [ B] E = E + E = q E+ v B total E B F = qv B B This defines B. F v, B F qvbsinθ direction by Right Hand Rule B is a vector field [ ] [ ][ ] = F N N T( tesla) q v = C m/ s = A m = 1 T = 10 4 gauss (earth magnetic field at surface is about 0.5 gauss) ( ) called Lorentz force For q > 0

9 Lecture 8-98 Physics 219 Question 1 February 6, An electron (charge e) is moving in +y direction where there is a magnetic field B in x direction. What is the direction of the magnetic force on the electron? a) along +x b) along -x c) along +y d) out of the plane (+z) e) into the plane (-z) v B y x

Lecture 8-108 Magnetic Force Does Not Do work!

10 Lecture Magnetic Force Does Not Do work! F = qv B -e F v Magnetic force only bends the direction of the moving charge and does not affect its speed. Magnetic force does not do any work.

Lecture 8-118 Bubble Chamber e - Since the curvature indicates that F m is in

11 Lecture Bubble Chamber e - Since the curvature indicates that F m is in opposite direction to v x B, the charge must be negative. v uniform B circular motion

Lecture 8-128 Charged Particle Entering Uniform Magnetic Field B qvb r v = m r mv = = qb 2 const.

12 Lecture Charged Particle Entering Uniform Magnetic Field B qvb r v = m r mv = = qb 2 const. Circular motion of charged particle q in homogeneous magnetic field. v B v qb f = = 2π r 2π m qb ω = 2π f = m Cyclotron frequency proportional to B proportional to q/m independent of v T 1 2π m = = Cyclotron f qb period

13 Lecture More complicated situations? v is not perpendicular to B Also non-uniform B magnetic bottle helical motion electron in magnetic field Van Allen belts

Electricity & Optics

Physics 241 Electricity & Optics Lecture 12 Chapter 25 sec. 6, 26 sec. 1 Fall 217 Semester Professor Koltick Circuits With Capacitors C Q = C V V = Q C + V R C, Q Kirchhoff s Loop Rule: V I R V = V I R