g, E and B vector addition

Since E field topic is so badly understood, why not do 3 in 1 .At a point X along the line joining this to circles.Let us find the g,E and B field at X.
g field:g = g1 + g2 ( draw g towards the mass at all times, one has to be negative as they are opp in direction )
E field : E = E1 + E2 ( draw E away from charge if positive and towards if negative at point X. Consider if they are the same direction or opposite to either add of subtract the E magnitudes)
B field B: B1 + B2 ( draw up or down from X depending the current is into page (CW) or out of page (ACW) . If both current point IN then at Point X B1 and B2 is up and down and therefore need to subtract.
All the above are just dealing with vectors in general and therefore resultant vector is always written as add but need to subtract if it is pointing in opposite direction.
The magnitudes have to be calculated separately using their respective formulas:\
g = GM/r2, E = kQ/r2 and B = kI/r2(not needed)

Particle accelerators

pic taken commons.wikimedia.org
Linear accelerator : Stanford linear accelerator aerial view in California , the longest in the world , length 3.2 km
Charged particle like electrons and positrons can be accelerated to a very fast speed inside a tunnel seen in picture.

Collisions due to charged particles

pic taken from onemansblog.com showing the effects of collisions of charged particles in the atmosphere near the poles known as aurora
Link to google search aurora borealis
Charged particles can be electrons , protons and ions definitely not neutrons. Therefore the neutrons are referred to as 'silent bullet' in mandarin as it will head towards its target and destruct the nucleus of cells. Electrons can easily be deflected due to its small mass. An electron heading towards and atom will most probably be deflected by electrons of the atom and also ionising or causing exitation of the atom . A proton moving towards an atom will also be deflected by the nucleus of the atom of a large charge of the same type thus the proton suffers greater acceleration due to its small mass. The force between two charges are the same ( action -reaction) but its acceleration on each charge is different due to differing masses. Relate this to the earth-moon system having the same force of attraction , but the moon accelerate in circular motion around the earth as the magnitude of the acceleration is greater.

Accelerating charged particles

Charged particles can be accelerated to a high velocity by applying an electric field in the direction of the velocity. Using energy equations, we can simplify by using potential difference V to find the velocity. This voltage will be called accelerating voltage V. Assuming the initial velocity to be nearly zero, the final v can be calculated from 1/2 mv2 = qV . For an electron the v would be ( 2eV/m)
If you look at the tube above, the orange heated element emits electrons that are accelerated forward by the blue anode (+) in two stages at different accelerating voltages. The fast beam of electron now pass through the 2 sets of yellow parallel plates , one set that moves the beam along the x- direction and the other set moves the beam along the y-direction. The pink cathode(-) controls the brightness by limiting the electrons from the heated element. The shape of the blue anode focus the beam on the screen. The yellow plates moves the beam along the x-y direction thus drawing the image at a very fast speed undetected by our eye.

pic taken from www.physics. sjsu.edu copyright addison and wesley , longman

Orthogonal field

link to addison and wesley longman

The two fields , E and B are at right angles to each other and at right angles to the velocity of the charge. With this arrangement the force due to E and B are opposite to each other . If the magnitude of the force is equal , there will not be any resultant force . The charge will be undeflected. The velocity of this charge v=E/B. ( FE = FB)

If many charges of different velocities enter this region , you therefore can select a desirable velocity by fixing up a value of E and B. This process is called velocity selection.

link to diagram

Combined fields

Combining three fields , g,E and B is really only E and B. The g field can hardly influenced the effect due to E and B. The acceleration due to g is only 9.81ms-2 as compared to acceleration due to E and B which is easily around 1 x 1014 ms-2. To sum up the effect of these three field is mainly to sum up the acceleration due to E and B bearing in mind their directions. Therefore need to do a vector summation using pythagoras theorem and tangent of angle.

Earth's magnetic field

The Earth's magnetic field shield us from radiation (charged particles) from outer space. If you see the Earth's field pattern , along the equator the field is parallel to the surface and at the poles the field is vertical. Charged particles spiral along the B field lines. The charged particles in the upper atmosphere will not spiral towards the surface around the equator but into the poles. Therefore , going to the poles can be a health hazard due to exposure to radiation. Jetsetting around the world is also a health hazard as we tend to come into the path of charges spiralling along the B field.

pic taken from www.vortexmaps.com

Motion in B field -advanced

link to : unsw.edu

If a charge enters a magnetic field and has a velocity component parallel to the B vector , the charge will continue to move with constant velocity. The circular motion is due the velocity component perpendicular to B .

The resultant motion is spiralling along B vector.

Comparing 3 force fields

pic taken from cmbi.ru.nl
Comparing 3 force fields:
The vector direction at a point away from source is
g - towards the mass
E - towards the charge if negative.
B - clockwise if current flow into page.
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Uniform(constant) fields exist
g- a few km apart
E - within parallel plates
B- inside a solenoid carrying current.
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Force is present
g- on a mass placed at that point in the direction of g
E - on a charge placed at that point in the direction of E if the charge is positive
B - on a moving charge placed at that point in the direction determined by Flemings LHR.

Uniform fields- motion

pic from web.ncf.ca

In this region, mass undergo constant acceleration due to gravity, g. Motion with constant g has been dealt with quite extensively in projectile motion.
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In this region, charges undergo constant acceleration a = F/m , where F = qE. Charges in E field behaves like a mass in g field where equations of motion can be applied.The difficulty is direction of acceleration in E field is not obvious as the gfield case.The acceleration is perpendicular the the length of the plates and the direciton is towards the negative plate.

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In this region, charges undergo acceleration a =Bqv/m which is perpendicular to the velocity. The resultant motion is circular and has the value a = v2/r . The Fleming's LHR will be help to determine the centre of the circle(direction of force) given the velocity of the charge and direction of Bfield.

Uniform fields-nature

We only assume g field to be uniform at distances that are close in gravitational terms , a few tens of km. g value is taken to be the same close to the surface of the Earth.
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Uniform E field is found between parallel plates of voltage V at a distance apart, where E = V/d.
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Uniform B field is found inside a solenoid carrying current. Its direction can be found by gripping the coil with right hand.

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B vector due to moving charge

Magnetic flux density, magnetic field strength or sometimes carelessly known as magnetic field refer to the B vector. B vector has the same treatment as E and g vector but somehow it is more difficult to visualize as it is 3D. Therefore to help us with 3D we employ your fingers to help us .

For a straight conductor carrying current- Use right hand and the thumb to point to the direction of current (velocity of positive charge) we curl the fingers to help us visualize the circular field with direction indicated by the fingers.

If two different currents flow near a point , then separately use right hand to find B1 and B2 and add them up as you would do to vectors.

Work done in E field

pic taken from sparkNotes, showing W = qV where V=Ed
Consider two points A and B , distance d apart along the E direction. Place a positive charge Q at A , if the force is towards B , then in moving from A to B , the charge will increase in velocity and we say the electric field does work on the charge. Work done QV= increase in KE . If the charge is moved to A from B, the force on the charge is still the same and work has to be done against this electric force. The formula remains very much the same, QV = increase in EPE = decrease in KE. Analogy: A stone falls from a height , workdone by the g field and the stone increase in ke. If you want the stone to rise up , there is loss in ke and this work done against weight is stored as gravitational potential energy. To handle a negative charge , be always aware of the force acting on it. The force on a negative charge is opposite to the E field vector. The force on a positive charge is in the same direction as the E field vector.

Force on charge

Force on a charge,F = QE, don't you think it looks similar to F= mg. Force is produced on a charge in the presence of electric field. Just as force is produced on a mass in the presence of gravitational field.In a constant electric field E, there will be constant acceleration of the charge , a = QE/m. In a constant g field(near the surface of Earth) , the constant acceleration a = F/m = g. If you can solve projectile questions well , you can do the same for a mass in g field or for a charge in E field. The difference being the acceleration of a charge is of a larger order of magnitude compared to a mass in g field.Typical values of g= 9.81ms-2 and a = 1.0 x 10(power 15 ) ms-2. Its acceleration and force are in the direction of E if a positive charge is placed within the plates where E is constant. If a negative charge is placed between the plates , E remain the same but the force is in the opposite direction and the acceleration is opposite to E .The equation of motion is applicable for both these cases of constant acceleration.

Energy conversion

See meteor shower on youtube
The surface of the Moon is full of craters as there is no atmosphere to protect itself from meteors. The loss of gravitational potential energy as an object falls from space must be converted to other forms of energy namely kinetic energy and heat energy. Atmosphere produces resistive force than in turn causes work to be done against it hence producing heat energy. This will limit the further increase in velocity as resistive force increase rapidly with velocity. The temperature of the object gets so high that the material just vaporises. By the time the object reaches the surface of the Earth it would have disappeared in the air , saving us from being hit . Now and again we do hear of falling objects from the sky but so far no fatality. The Moon is not so fortunate and therefore its surface is marked with craters. See link to the map of the Moon

Electric field intensity

pic taken from sparknotes , for clear diagrams-go to link
Electric field strength , electric field vector all mean the same the E -vector . If you have the right concepts in g- field , you can have an easier time here. Just as a mass has g- field around it , a charge has E- field around it. The formula is different for E field , E=kQ/r2 where k = 1.11x10-10(4phi ephsilon 0) but it is similar to g where it is inversely proportional to square of distance away from the charge.
The direction is away from the charge from the point concerned if the charge is positive . If the charge is negative , draw the E vector pointing towards the charge from the point concerned. In E field you have to be more cautious about the directions.
So with magnitude and direction described above you can calculate the vector E a distance away from a point charge. If there is more than one charge , then there will be more than one E vector , E1 due to charge one and E2 due to charge two. Using vectors addition you can now find the resultant vector at the point concerned. You can use component method followed by pythagoras theorem and tan (angle) to find the resultant E-vector .
pic taken from zebu.uoregon.edu