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  • Revision:Electricity and Magnetism 2

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12.1 Electricity force, field and potential

12.1.1 : Coulomb's Law

\displaystyle F = \frac{Q_1Q_2}{4 \pi  \epsilon_o  r^2}

The force is dependent on the two changes and the square of the distance between them. Opposites attract and like repel. Vector addition of these forces may be necessary. The force on each of the two charges in question is opposite, but equal in magnitude. Q is in coulombs, r in meters.

12.1.2 : Electric field strength is equal to Force/charge

Therefore, to find the field strength at a given point, put a point charge of 1 coulomb at the point, and then calculate the force on it, and thus find the field strength and direction by vector addition.


12.1.3 : Electrostatic potential

Defined as the work done (in joules) to move a charge from infinity to a given point in the field, over the charge (in coulombs), or:

\displaystyle V = \frac{W}{q}.

The SI unit is volts (where a volt is a joule per coulomb). This value is a scalar, and so it doesn't matter what path you take between two points. In a radial filed:

\displaystyle V = \frac{1}{4\pi \epsilon_o} \frac{q}{r}.

As long as r is outside the conductor producing the field (if it's inside, then V = 0. In this case, q is the charge on the point creating the field. (both equations are in the data book)

12.1.4

1 electron volt is defined as the work done to move an electron through a potential difference of 1v. This can be plugged into the above equation, to show that 1eV = 1.6 x 10-19J.


12.1.5

The potential at a point in a field around a point charge or a uniformly charged sphere can be found with the above equation:

\displaystyle V = \frac{1}{4\pi \epsilon_o} \frac{q}{r}.

For a collection of point charges, the fact that electric potential is a scalar greatly simplifies the problem, because direction need not be considered. Simply add up all the different potentials from each point charge to find the total. Inside a hollow sphere, the V will be zero until you're outside the sphere, then jump to a max, then fall away to zero at infinity.

12.1.6

For parallel plates, the equipotential lines run parallel between the plates, and then diverge out from either end. With point charges, two opposite charges create circles of equipotential around them, but squashed in on the side closes to the other point...eventually there will be a line running straight between them. (Like charges don't seem to be necessary, but) Like charges will produce a sort of figure 8 with the center cut out, and each point charge in one of the 'holes' there are a bunch of these figure 8 shapes radiating out. Between parallel plates, equipotential lines can be related to electric field strengths, because the evenly divide up the space between the plates, as does the field strength. Also, equipotential lines will always cross filed lines at 90o, so that should help find exactly where they go.


12.2 Magnetic fields

12.2.1

Find the force on a moving charge, or an electric current in a magnetic field...


Moving charge - using the right and palm thing, find the direction of the force (remembering we're talking about conventional current, so it goes forward for +ve charges, backward for negative). Find the acute angle between the direction of the magnetic field at that point, and the direction in which the particle is traveling. Sub into:

\displaystyle F = qVB\sin \phi to find the magnitude of the force (\phi is the angle, B is the field strength, V is the velocity and q is the charge).


Current carrying wire - find the direction of current flow in the wire, and thus the direction of the force. Find the angle between the field lines and current flow. Sub into:

\displaystyle F = IlB\sin\phi to find the magnitude.

Both of these equations are in the data book. When \phi = 0^o, ie when the motion is parallel to the field, there will be no force.


12.2.2 : Some devices

Galvanometer - A permanent magnet is set up around a loop of wire. This wire is allowed to rotate on a axis, but has a spring attached to always pull it pack to parallel when there is no force. When a current is passed through the loop, this causes a force on the loop, and it rotates. This pivoting moves a marker attached to the axis, and so shows the current which is flowing through, since the grater the current, the further it will turn before the force is equalized by the spring.


Loud speaker - A loud speaker consists of a metal coil attached to the stiffened cardboard of the speaker, with a permanent magnet surrounding the coil. When there is current running through the coil, this causes forces back and forth (as appropriate) on the coil, and thus it vibrates the cardboard. Therefore, with the right alternating current, different frequency vibrations can be produced, thus producing sound.


Electromagnetic relay - he object of a relay is to complete a circuit when a current is passed through an electromagnetic. This is done with a coil of wire which, when current is passed through it, pulls a piece of metal near it towards it. Since this metal is on a pivot, it forces a wire attached to it above up, and onto the other wire, thus completing the circuit. When the current in the coil is released, the spring action disconnects the circuit.


12.2.3

The magnetic field around a current carrying wire is defined by:

\displaystyle B = \frac{\mu _o}{2 \pi} \times \frac{I}{r}.

I is the current through the wire, and r is the distance away from the wire where the field is being measured.


For a solenoid at least 10 time as long as it is wide, the field inside is constant, and defined by:

\displaystyle B = mu _0 \frac{NI}{l}.

where N is the number of loops of wire, l is the length, I is the current and \mu _0 is a constant. The field strength is equal anywhere inside the coil. (both equations are given in the data book)


12.2.4

The force between two parallel conductors is defined by:

\displaystyle F = \frac{\mu _0}{2 \pi} \times \frac{I_1I_2l}{r}.

where r is the distance between them, I_1 and I_2 are the currents in each and l is the length of both wires. Force per meter can be found simply by subbing in 1. As can be seen, if the currents are in opposite directions, the force will be negative, or away from the other conductor.


The ampere is defined based on this (the current which produces a force of 2 x 10-7N between two wires 1 meter apart), since it is easy to accurately vary the current in a wire. The coulomb is then defined from this (charge in coulombs = current x time).


12.3 Electromagnetic induction

12.3.1

Magnetic flux is defined as BA, or field strength x area...it's symbol is \phi. If the field is not perpendicular to the area in question, however, it's BA\cos\phi, where \phi is the acute angle between the field direction and the normal to the area. Flux linkage is basically the change in N\phi, where N is the number of turns of wire, and \phi is the flux.


12.3.2

Neumann's equation is:

\displaystyle  E = -N \frac{\Delta \phi}{\Delta t}

or the induced emf is equal to the number of loops x the change in flux over time.

Lenz's law says basically that an induced emf will always produce a current who's magnetic field opposes the original change in flux. Farraday's law is that the induced emf is proportional to the flux cut divided by the time taken.


12.3.3

\mathrm{emf} = Blv

This is for a wire of length l moving perpendicularly through a field of strength B at a velocity v.


This can be derived from \displaystyle F = \frac{\Delta Ø}{\Delta t} as follows:


\displaystyle F = B \frac{\Delta A}{\Delta t} = B l v \frac{\Delta t}{\Delta t} = Blv.


Or from F = IlB

  • charge - Q
  • voltage - V
  • work done - W

\displaystyle QV = W

\displaystyle V = \frac{W}{Q} = \mathrm{Force} \times \frac{\mathrm{distance}}{Q}

\displaystyle = IlB \times \frac{s}{Q} = \frac{(\frac{Q}{t})lBs}{Q}.

The Qs cancel out and we're left with:\displaystyle V = lB (\frac{s}{t})=Blv.

12.3.4

I assume what we're talking about here is a loop of wire rotating in a magnetic field ...

We take a coil with N turns of wire, and an area A rotating in a field of strength B at an angular velocity of \omega. The flux linkage at any given point is: BAN \sin (\omega t) , where t is the time, as it rotates, starting form to when the entire coil is parallel to the field, and so produces no emf).

We then play with some calculus (sub it into neumann's equation, N being already accounted for, and do it like a calculus equation) to get \mathrm{emf} = BANw \cos (\omega t}.This equation is not in the data book.


12.3.5

An AC generator is, just like above, a coil being force to rotate in a magnetic field. This produces an alternating current because each half turn, the effective orientation of the coil is reversed...the side that was going left is going right, and so the current is also reversed. The two ends of the loop are connected to slip rings with are allowed to turn, and brushes rubbing on them, and running the alternating current out to the rest of the circuit.


12.3.6

Average power consumption = I_{rms}V_{rms} and I_{rms} = \frac{I_0}{\sqrt{2}}. Same for V_{rms}. This equation is in the data book, and can be derived as follows:

\displaystyle P = I_0V_0 x \sin (2\omega t)

Thus P_{av} = \frac{1}{2}I_0V_0 =

and so the above rms bits can be found. V_{rms} = \frac{V_0}{sqrt{2}}.


12.4 The Cathode ray oscilloscope (CRO)

12.4.1

A COR is basically a tool for measuring variations in current from a source. The CRO provides a continual graph of the current over time on the screen...it's difficult to describe how exactly to use one...you'll have to have tried it.


How it works -- basically, at the back there is an electron 'gun' where a beam of electrons are produced by a large potential difference between an anode and cathode (the 'ray' comes off the cathode, making it a cathode ray). It the passes through two sets of perpendicular deflectors. The horizontal one is controlled by the settings of the CRO, and makes the light trace from left to right. The vertical one is indirectly controlled by the source current, and produces the up and down sine type curves.


The differences between this and a TV tube is that a TV strikes every pixel with the cathode once per frame (and may have multiple pixels with different colors) The CRO, however, strikes in a sine curve through each pass ... not much else to say really, they both work on the same principle.


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