RC Circuits: Discharging

RC Circuits RC Circuits: Discharging ▪ ▪ ▪ ▪ The figure shows a charged capacitor, a switch, and a resistor At t = 0, the switch closes and the capacitor begins to discharge through the resistor The second figure shows the circuit after the switch was closed Kirchhoff’s loop law applied to this circuit clockwise is 𝑄 o ∆𝑉𝑐𝑎𝑝 + ∆𝑉𝑟𝑒𝑠 = − 𝐼𝑅 = 0 ▪ ▪ Q and I are instantaneous; they keep changing The resistor current is the rate at which charge is removed from the capacitor: 𝑑𝑄 o 𝐼 = − 𝑑𝑡 𝐶 RC Circuits: All results for discharging ▪ The charge on the capacitor of an RC circuit is 𝑄 = 𝑄0 𝑒 −𝑡/𝜏 where 𝑄0 is the charge at t = 0, and 𝜏 = RC is the time constant ▪ The capacitor voltage is directly proportional to the charge, so ∆𝑉𝐶 = ∆𝑉0 𝑒 −𝑡/𝜏 where ∆𝑉0 is the voltage at t = 0 RC Circuits: Charging ▪ ▪ This circuit charges a capacitor The capacitor charge and the circuit current at time t are 𝑡 𝑄 = 𝑄0 (1 − 𝑒 −𝜏 ) 𝐼 = 𝐼0 𝑒 −𝑡/𝜏 where 𝐼0 = ε/R and 𝜏 = RC Magnetism ▪ ▪ ▪ ▪ Tape a bar magnet to a piece of cork and allow it to float in a dish of water It always turns to align itself in an approximate north-south direction The end of a magnet that points north is called the north-seeking pole The end of the a magnet that points south is called a south pole ▪ If the north pole of one magnet is brought near the north pole of another magnet, they repel each other ▪ ▪ Two south poles also repel each other The north pole of one magnet exerts an attractive force on the south pole of another magnet ▪ ▪ The north pole of a bar magnet attracts one end of a compass needle and repels the other Apparently the compass needle itself is a little bar magnet ▪ ▪ ▪ Magnets can pick up some objects, such as paper clips but not all If an object is attracted to one end of a magnet, it is also attracted to the other end Most materials, including copper (a penny), aluminum, glass, and plastic, experience no force from a magnet Magnetism – basic facts ▪ ▪ ▪ ▪ Magnetism is not the same as electricity Magnetism is a long-range force All magnets have two poles, called north and south poles. Like poles repel; opposite poles attract Materials that are attracted to a magnet are called magnetic materials. The most common magnetic materials are iron and steel Electric Current causes a magnetic field ▪ The right-hand rule determines the orientation of the compass needles to the direction of the current Notation for vectors and currents perpendicular to the page ▪ Magnetism requires a three-dimensional perspective, but two-dimensional figures are easier to draw Magnetic fields from moving charges Source of the Magnetic Field: Moving Charges ▪ The magnetic field of a charged particle q moving with velocity v is given by the BiotSavart law: 𝜇 𝑞𝑣𝑠𝑖𝑛𝜃 o 𝐵𝑝𝑜𝑖𝑛𝑡 𝑐ℎ𝑎𝑟𝑔𝑒 = (4𝜋0 𝑟2 ) direction given by the right-hand rule Calculating the Magnetic Field ▪ ▪ The constant 𝜇0 in the Biot-Savart law is called the permeability constant: o 𝜇0 = 4π x 10-7 T m/A = 1.257 x 10-6 T m/A The SI unit of magnetic field strength is the tesla, abbreviated as T: o 1 tesla = 1 T = 1 N/A m Magnetic Field of a moving positive charge ▪ ▪ The right-hand rule for a moving positive charge is similar to the rule for a currentcarrying wire The component of B parallel to the motion is zero Biot-Savart, now with cross product ▪ The magnetic field of a charged particle q moving with velocity v is given by the BiotSavart law: 𝜇 𝑞𝑣𝑟 o 𝐵𝑝𝑜𝑖𝑛𝑡 𝑐ℎ𝑎𝑟𝑔𝑒 = (4𝜋0 𝑟2 ) magnetic field of a point charge The magnetic field of a current-carrying wire ▪ ▪ Current = many moving point charges The magnetic field of a long, straight wire carrying current I at a distance r from the wire is 𝜇 𝐼 o 𝐵𝑤𝑖𝑟𝑒 = (4𝜋0 𝑟) long, straight wire