IB Problem Set 17:  1
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5  7  11  15
 17

19  21  AP#4  Go up
 by Will Gannett, class of 2002, IB Physics HL
1. How much work is needed to move a 8.6mC charge from ground to a point whose potential is +75V?
Well, I know that the equation to use is W=qV, where W is work in joules, q is charge in Coulombs (sp?), and V is voltage in Volts. q is obviously 8.6mC (8.6E6 C), and V will be 75, since it has to go from the ground (0V) to the point (75V). Since the charge is negative and it is being moved from 0V to 75V, we know that any work done in this case will not be work required from an external source, but work done by the field itself (in the form of electrostatic potential energy being converted to kinetic energy). The amount of work done will be W=qV=8.6E6*75=6.45E4J. Again, this in negative to show that it is work done by the field rather than work required.
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3. How much kinetic energy will an electron gain (in joules and eV) if it falls through a potential difference of 21,000V in a TV picture tube?
There are two ways to do this problem. One will give you the answer in eV, and one will give you the answer in joules. Neither are difficult. The first way is the more complicated, and uses the formula from the previous question, W=qV. V=21000, the potential difference, and q is the charge of one electron, or 1.602E19C. Therefore, the work (which will all be converted to kinetic energy) is 21000*1.602e19=3.4E15J. To get the answer in eV is much easier. We know that it is one electron being accelerated through 21000V. Therefore, the total number of electron Volts will be 21000, or 21keV.
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5. How strong is the electric field between two parallel plates 5.2mm apart if the potential difference between them is 220V?
Well, you should know that there are two different units for electric field strength: either N/C, or V/m. These are both in the data packet, as E=F/q and E=DV/Dx. E=DV/Dx is obviously the one to use here.
E=DV/Dx
E=220/5.2E3
E=42000V/m
The sign of the answer only shows the direction of the field.
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7. What potential difference is needed to give a helium nucleus (Q=2e) 65.0keV of KE?
We know that a single electron, moved through V volts, will gain V electron volts of energy. Similarly, 2 electrons (or an object with the same charge as two electrons, such as a helium nucleus) will gain 2V electron volts of energy when moved through V volts. From this, we can see that an object with a charge of two electrons (or protons) that has 65000eV of energy would have to only be accelerated through half the voltage a single electron would have been. This means that the answer is 65000/2, or 32500V. To do the math:
65000eV/2e=32500V.
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11. What is the speed of a proton whose kinetic energy is 28.0 MeV?
I think that the easiest way to do this would be to use the equation W=.5*m*v^{2}. First, we must convert 28.0MeV to joules. This is a simple matter.
28.0E6eV*1.602e19C/e
=4.49E12VC
=4.49E12J
So, the proton has 4.49E12 joules of kinetic energy. Since we know the mass of the proton (1.67e27kg, for those of you too lazy to actually look at your data packet), it is another simple matter to solve for the velocity.
W=.5*m*v^{2 }4.49E12=.5*1.67E27*v^{2 }4.49E12/.5/1.67E27=v^{2 }v^{2}=5.38E15
SO, v=7.33E7ms^{1}
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15. A +30 micro coulomb charge is placed 32 cm from an identical +30 micro coulomb charge. How much work would be required to move a +0.50 micro coulomb charge from a point midway between them to t point 10 cm closer to either of the two charges?
All we need to do is multiply the test charge by the change in voltage. V = W/q, so the work, W = Vq where V is the change in voltage, and q is the charge moved.
We now have two locations, one directly between the two +30 micro coulomb charges, that is 16 cm from each of them, since they are 32 cm apart, and a second location that is 6 cm from one of the charges, and 26 cm from the other. See the diagram below:
So to solve these problems you just do these three steps:
1. Find the voltage before and after the move
2. Find the difference in voltage (subtract)
3. Use W = Vq to calculate the work done
So the voltage before is simply V = kq/r + kq/r (yay no vectors)
Vbefore = 8.99e9(30e6)/.16 + 8.99e9(30e6)/.16 = 3371250 Vand after it is: (note the distances are the only things that change)
Vafter = 8.99e9(30e6)/.06 + 8.99e9(30e6)/.26 = 5532307.692 VThe change is that we went up in voltage from 3371250 V to 5532307.692 V, a change of 2161057.692 volts.
Since we are moving a charge of 0.50 micro coulombs through this difference, the work is simply
W = Vq, W = (2161057.692 V)(0.50e6 C) = 1.08 J
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17. How much voltage must be used to accelerate a proton (radius 1.2 x 10^{15} m) so that it has sufficient energy to penetrate a silicon nucleus? A silicon nucleus has a charge of +14e and its radius is about 3.6 x 10^{15} m. Assume the potential is that for point charges.
Well, let's first say that the nucleus is stationary and the proton free. We know that the point at which it will begin to penetrate is equal to the radius of the silicon nucleus plus the radius of proton. This is 1.2E15+3.6E15, which is 4.8E15m. Since the proton is only able to get this close to the nucleus from kinetic energy, it makes sense that whatever potential energy it now has due to proximity must have been obtained from the voltage potential. Since the charge on the proton remains the same and it has no kinetic energy both at the beginning and the end (it starts out at rest and ends at rest, due to the voltage potential from the charge of the nucleus), the voltage at those points must be the same (i.e. the voltage difference from the start to the point of penetration is zero). What we want to find is the voltage at that point, which will be the same as the voltage potential through which the proton was accelerated. To do this, we use the equation V=kq/r.
V=8.99e9*14*1.602E19/4.8E15
V=4.2E6=4.2MV
You will notice that the mass and charge of the particle are not relevant. This implies that this would work with alpha particles as well, except for the fact that the radius would be different. However, an alpha particle accelerated through that voltage would travel to the same distance from the silicon nucleus.
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19. Consider point a which is 70 cm north of a 3.8mC point charge, and point b which is 80 cm west of the charge (Fig. 1723 in Giancoli). Determine (a) V_{ba} = V_{b}  V_{a} and (b) E_{b}  E_{a} (magnitude and direction).
(a) Ok, first we need to find V_{b} and V_{a} using V=kq/r
V_{b}=8.99E9*3.8E6/0.8=42703V
V_{a}=8.99E9*3.8E6/0.7=48803V
So, V_{ba}=4270348803=6100V
(b) The electric field is a bit more complicated. We will use a vector at each point to represent the electric field strength there and calculate the field strength using E=kq/r^{2}. At point b, E=8.99E9*3.8E6/0.8^{2}=53378N/C. Since a positive charge placed at point b will travel to the right (it is to the left of a negative charge), we know that the vector will only have an xcomponent, and it will be positive. So, the field at that point is represented by 53378x+0y. The point b is similar. E=8.99E9*3.8E6/0.7^{2}=69718N/C. The vector at that point will only have a negative ycomponent, so the vector will be 0x+69718y.
Moving to the next step, E_{b}E_{a} = (533780)x+(069718)y=53378x+69718y, in vector component form. (Isn't this great? It's just a chapter 2 problem in disguise!!) The question asks for the answer in anglemagnitude form, though. So, we need to convert. The magnitude is easy. It's just the Pythagorean theorem. 53378^{2}+69718^{2}=c^{2}, so c=87806N/C. As for the direction, tan^{1}(69718/53378)=53º from vertical, up and to the right. So in conclusion, the answer is 8.8E4N/C, 53° E of N.
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21. Two identical +7.5mC? point charges are initially spaced 5.5 cm from each other. If they are released at the same instant from rest, how fast will they be moving when they are very far away from each others? Assume they have identical masses of 1.0 mg.
To calculate this, we will use electrostatic potential energy.
Now, we know that E_{pot}=E_{kin}, because energy is conserved, and the potential energy of the system will equal the amount of work required to put the charges in their current state. We also know that V=kq/r and V=W/q. So, since kq/r=W/q, W must equal kq^{2}/r.
W=kq^{2}/r
W=8.99E9*(7.5E6)^{2}/0.055
W=9.194J
So the total energy of the system is 9.194 joules. Since both charges have the same mass, and momentum is conserved, they will have equal and opposite velocities. To find these, we use good old 1/2*m*v^{2}. Also, because they have the same mass, they will end up with the same amount of energy, 9.194/2, or 4.597J.
1/2*1E6*v^{2}=4.597^{ }v^{2}=4.597*2/1E6
v^{2}=9.194E6
v=3.0E3m/s
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<solution goes here>
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