Electric Fields • Electric Charge Properties • Opposite signs attract, same signs repel • Charges are quantized and they are conserved in isolated systems • Coulomb’s Law defines the electric force felt or applied by one charge to another • Electric fields defines the force felt per unit charge at some point in space • The electric field generated by multiple charges is simply the vector sum of all the individual electric fields at some point
Electric Fields • The total electric field at point P is going to be the sum of the two electric fields • Since electric fields are vectors we need to use vector decomposition to find the x component of the electric field and y component of the electric field • The electric field vector of • The electric field vector of
Electric Fields • The net electric field in the x direction • The net electric field in the ydirection
Gauss’s Law • Electric Flux is defined as number of electric field lines that pass through a given surface • The angle is the angle between the area vector (which will be normal to the surface of the area) and the electric field vector • Gauss’s Law says that the net electric flux through any closed GaussianSurface is equal to the net change inside divided by • The electric field inside a conductor is zero. All the charges are located on the surface of the conductor.
Gauss’s Law • Because of the symmetry of the electric field we can use Gauss’s Law to find the electric field at some point r • From the image we can see that a cylindrical Gaussian Surface will be the best choice. • This is because the angle between the area vector and electric field vectors is zero • Applying Gauss’s Law
Gauss’s Law • is given by λ multiplied by the length of our Gaussian Surface • Lastly the coulomb constant (k) can be introduced
Electric Potential • Moving a positive charge through an electric field changes the charge-field system’s potential energy • The electric potential or voltage of a system is defined as the change in potential energy per unit charge • Note that voltage is a scalar value not a vector • Based off the above definition we can also define the electric field as the negative derivative of the voltage • Similar to electric fields the net voltage as some point in space can be found by summing together all the individual voltages together
Electric Potential • Since voltage is a scalar value, no vector decomposition is needed. Thus we just add the two voltages together
Capacitance • A capacitor is made of two conducting plates that store charges when a voltage is applied. • The capacitance of any capacitor is given by the following equation: • Capacitance only depends on the geometry of the capacitor • In series the equivalent capacitance is given by: • In parallel the equivalent capacitance is given by:
Capacitance • When ever we are asked to find the equivalent capacitance of a system, we essentially want to combine all the individual capacitors into a single one • The junction after goes to both . Because of this we can say that are in parallel and that their equivalent resistance is in series with • parallel • in series with
Current and Resistance • Current is defined as the amount of charge that passes through a cross sectional area in some time interval • Resistance is the inverse of the conductivity of a conduct and is a geometric value: • Ohm’s Law relates the potential difference across a conductor to its resistance and how much current is passing through • If there is a potential difference across a given element than energy is being delivered to said element. The rate at which the energy is given is the power:
Current and Resistance • An electromotive force or emf is voltage across the battery • The equivalent resistance of resistors in series is given by: • The equivalent resistance of resistors in parallel is given by: • Kirchhoff’s Rules • Junction Rule: at any junction, the sum of the currents must equal to zero • Loop Rule: The sum of the potential differences across all electric elements around any circuit loop is zero.
Current and Resistance • Since the resistors are in parallel the equivalent resistance is given by: • In parallel the voltage across each resistor will be the same. Thus Ohm’s Law can be used to find the current • Since we know the current across each resistor we can easily find the power delivered to each resistor
Magnetic Fields • If a charged particle travels with some velocity through an external magnetic field is will experience a magnetic force given by the following equation: • Or if a wire carrying a current is placed in a magnetic field • Magnetic forces do not do any work on moving particles • However, magnetic forces can change the trajectory of the particles path
Magnetic Fields A proton is moving in a circular orbit of radius 0.14 meters in a uniform 0.35-T magnetic field perpendicular to the velocity of the proton. Find the speed of the proton. • The magnitude of the magnetic force felt by the proton • In this case the magnetic force is causing the centripetal force • Solving for velocity:
Sources of Magnetic Fields • The Biot-Savart law says that the magnetic field at some point in space is due to length element that carries a steady current • Ampere’s Law says that the line integral of the magnetic field around any closed path ( in this case this closed path is called an Amperian Loop) is equal to the total steady current through the closed path • Both of these law show that there is a relationship between currents and magnetic fields • Gauss’s Law of magnetism: • The magnetic field equations for both a toroid and solenoid
Sources of Magnetic Fields A long, straight wire of radius R carries a steady current I that is uniformly distributed through the cross section of the wire. Calculate the magnetic field at a distance r from the center of the wire in both
Sources of Magnetic Fields • Ampere’s Law
Faraday’s Law • Magnetic Flux is defined as the “amount” of magnetic field passing though an area: • Faraday’s Law of induction states that the emf induced in a loop is directly proportional to the time rate of change of magnetic flux through the loop: • Or in its general form • Len’s Law states that the induced current and induced emf occurs in the opposite direction as to create a magnetic field which opposes the one that produced them
Faraday’s Law • The conducting bar moves on two frictionless, parallel rails in the presence of a uniform magnetic field directed into the page. The bar has a mass m, and its length is l. The bar is given an initial velocity v to the right and is released at t=0. • Part A: Find the velocity of the bar as a function of time. • Part B: Show that the same result is found by using an energy approach.
Faraday’s Law • As the bar moves there is a change in the magnetic flux • Faraday’s Law • Newton’s Law to find velocity
Inductance • When a current is a loop of wire changes with time an emf will be produced as a result of Faraday’s Law • This is because a current that is changing in time will cause the magnetic flux to change in time • Inductance is a measurement of how much opposition the loop offers to a change in the current of the loop. • Similar to capacitance its value is based upon the geometry of the inductor • The inductance of a solenoid: • Mutual Inductance allows us to relate the induced emf in a coil to the changing source current in a nearby coil:
Inductance • Consider a uniformly wound solenoid having N turns and a length L. Assume L is much longer than the radius of the windings and the core of the solenoid is air. • Part A: Find the inductance of the solenoid • **Remember inductance is the L value**
Inductance • Before the current is turned on there is no magnetic field in the solenoid. But when the current is turned a magnetic field will be created over some time interval • A changing magnetic field means there is a changing magnetic flux • Since there are N turns we have • From Faraday’s Law • Thus the inductance of the system is:
Alternating Current • RMS Current and Voltage • If an AC circuit consists of only a resistor and a source, then the current is in phase with the voltage • If an AC circuit consists of only a inductor and a source, then the current lags the voltage • If an AC circuit consists of a capacitor and a source, then the current leads the voltage
Alternating Current • The Impedance of an RLC Circuit: • The angle between the applied voltage vector and current is called the phase angle: • The RMS Current for RLC Circuit: • Power of an AC Circuit • The maximum rms current value occurs at the resonance frequency. This is the frequency at which
Alternating Current • A: Find the inductive reactance, capacitive reactance, and the impedance. • B: Find the maximum current in the circuit. • C: Find the phase angle between the current and voltage. • D: Find the maximum voltage across each element.
Alternating Current • Find the inductive reactance, capacitive reactance, and the impedance. • Find angular frequency • Find inductive reactance • Find inductive capacitance • Find impedance
Alternating Current • Find the maximum current in the circuit • Find the phase angle between the current and voltage • Find the maximum voltage across each element
Electromagnetic Waves • Lorentz Force describes the force felt by a charged particle in the presence of both an electric and magnetic field: • Maxwell’s Equations • Electromagnetic waves are predicted by Maxwell’s equations. • Speed of Light
Electromagnetic Waves • The relationship between the frequency and the wavelength of an electromagnetic wave: • The electric field and magnetic field are perpendicular to each other as well as the direction of propagation • The instantaneous magnitudes of E and B is given by the following equation: • Electromagnetic waves carry both energy and momentum • The Poynting vector represents the rate at which energy passes through a unit area by electromagnetic radiation ( in other words the power per unit area) • Wave Intensity
Electromagnetic Waves • Start with the relationship between frequency and wavelength: • Use the relationship between frequency and period:
Electromagnetic Waves • To find the magnitude use the relationship between E and B: • c • Since the wave is traveling in the x direction and the electric field is in the y direction, the magnetic field must be in the z direction. This is because the fields are perpendicular to each other as well as the direction of propagation.