5

5_ If we have an electrically charged disk of radius R with surface density charge is given byKy? then the tota)CVR-z? Ky? dydx ~RJ-VRE-rQ = JI o6s,5 y)dACalculate ...

Question

5_ If we have an electrically charged disk of radius R with surface density charge is given byKy? then the tota)CVR-z? Ky? dydx ~RJ-VRE-rQ = JI o6s,5 y)dACalculate using polar coordinates.

5_ If we have an electrically charged disk of radius R with surface density charge is given by Ky? then the tota) CVR-z? Ky? dydx ~RJ-VRE-r Q = JI o6s,5 y)dA Calculate using polar coordinates.



Answers

A circular disk of radius $R$ and total charge $Q$ has the charge distributed with surface charge density $\eta=c r,$ where $c$ is a constant. Find an expression for the electric potential at distance $z$ on the axis of the disk. Your expression should include $R$ and $Q$ but not $c$

Ah, we choose a Gaussian surface of radius r, which is less than are not in concentrate Witty spear we calculated of electrically losing goes a lot black. The judge density the row e and it is given us row is a call to take you up on TV. Given that the charge and it is a function of the radial position, there is a role he R is equal to Rho or not are upon are not so the charge enclosed is equal to limit low e dizzy So we have zero to r ro, not hi dash upon are not or by our dish square the our dish. So Cuban closes or or not I r to the power for upon her or not Take this as equation one. This equation refers to the charge and closed So now, to find a total child assume eyes equal to earn no are not and the charge and city can be calculated using the formula Q role by are not to the power for upon are not so. That is a role not our but it will not buy are not cube so raw not will be queue up on by are not cube this situation two. So now substitute substitute the value of right or not from equation too. Immigration one we get you enclosed R is equal to a row, not by artists Power four upon raw notes Oh, that is a girl took you are upon are not to the power for, um Does the electric field everywhere with thin does feel is found using well this loss Now the the air is equal to Q and closed upon Absalon Not so that is the for by hi square is equal to Q Upon absolute note, Q Q are not to be power for but this implies he is equal to kill upon for by epsilon not I square upon are not to the power for Does the equation for electric field within the Yeah, in terms of kill are not an I And since the Josh distribution is positive, the electric field blind radial e outwards

Okay, So in this problem, we have Ah, we have to calculate the electric feuds at a distance. R from the access of a cylindrical charge distribution given that the position are were calculated is smaller, then their radios off the cylinder. Therefore, first of all, let's represent this cylinder here. We have cylinder represented here, and we're going to draw a Gulshan cylinder in sight. This cylinder, therefore, we have a smaller cylinder here, Okay? And because of that, we have are greater. Radios are and here and we have, ah, radios off the Gulshan cylinder, which is the We are capito, R and R. Okay, uh, what else do we know? We have the density off the cylinder of the charge density. Sorry. Okay. Uh, so since we choose a Goshen cylinder inside, they regional cylinder Ah, we're going to use to calculate the electric electric future. We're going to use the definition off electric forks in the go slow, Which is the clothes integral off the electric feuds. G A. And this is also equal the charge. The charge inside big ocean surface divided by absolute zero since the electric field to spiral out to the area because we can see from Let's put here in Read that the electric fuge is going always to be If you diss positive density actually is always going to be perpendicular to the guy ocean surface. Therefore, we can say that this close integral is all is just going to be Yes, Close Inter girl off the area. Okay, so and justice in terror is going to be recruited. Charge inside, divided by absolute here. No, If we calculate the surface area off the cylinder, we will find that this is going to be the electric field we went to Goku it The multiply ah, to buy to buy R l, which is the lateral air. The lateral surface area off the ocean cylinder. Okay, so this is equal the charge inside and we can see that the charge inside is going to be not the entire charge, But did then city multiply by the volume inside the Goshen surface. If I did by absolute zero, therefore, the volume off the Goshen surface is going to be hope. Hi. Are square l divided by absolute zero. Because this is the definition of volume is the area which is a circle with radios are multiply by the length off the the cylinder. Therefore, we can simplify and say that the electric feud generated by this, uh, cylindrical distribution of charge it's going to be Let me see Hope pie are to l divided by two pi r l absolute zero. Therefore, we can cross radios length by and finally or answer to this electric field is going to be hope, which is the density that multiplies the radios off the Goshen cylinder divided by true, absolutely zero. So we just need to use the definition off the Ghoshal flux in calculate with the ocean surface inside the original senator. That's all. Thanks for watching.

In this question. You have disk off regions are mhm. Then, uh, you're interested if I the electric potential appointee PCR he's a distance X away from the center. Yeah, you're interested to find the electric potential, and then the ring has a non uniform charge. Distribution. Uh, surface charge. Density. CR me. Damage. This radios are okay. Yes. So we want to find a electric potential at P. Okay. Okay. So to solve this problem, um, you are going to view the disk. Um, you got this? Yes, me off many concentric rings. Okay, so each concentric ring will have a charge D Q as a charge. Que que? Yeah. So So it's like So this is like the disk. And then from the center, you're going to you. We're going to cut the this into many small rings. Okay? Yeah. And so Okay, so he has a charged que it is like you. It goes to sigma two pi r p r uh, Sigmund the A and then the d A is sigma. The idea is to priority are Okay, so and then you put in our signal B c r plans to buy our We are okay. Okay, then, um, from the textbook. You know that for ring Okay for being bartering. Hey, electric potential. Hmm. A boring Or church B Q Thank you. Or remark Charge. All right, que Right, then. Uniformly charge. And then we have the electoral potential At T Okay. Ah, great. Yes. Are So this is what we have. Who just considering and radios are. Yeah, And then a point, he descends X away from the center. He the electric potential. We ring. Yes. Um e que que divided by the distance, which is our square plus x square. Uh huh. So if we have a small rings, okay, for each small ring in the case off the disk. Okay, kids, off the disk. Okay. We have TV ring or D V disc. Actually, if you go to que, uh, by square plus x square. Mhm. So we need to integrate. Um, so you re disk integrate both site, right? Can you? Did you divide by square root are square plus x square. Okay. And then we substitute our like Okay, we call that curious on c are too high. Are we are okay. So, k, you bring out the constant You're angry to integrate from zero to We are You can see times two pi r square. We are. Okay. Where are square? That's X square. Okay, so we continue to bring out constant. Okay? The pie. Yeah. Okay. So I'm going to have going to write like this are in to our square root are square packed x square. We are because we are going to do integration by parts. Okay, so So I'm going to, um, differentiate this. This is to be differentiated, and then this is to be integrated. Okay, Differentiate. Okay. To be you've been shaded, and then this time is to be integrated. Okay, Right. So continue E c high. Okay, so if we need to, uh so the first time put our and then the second term, that's to be integrated when we integrate this. Okay, we get to times square. Roots are square plus x square. Okay. So just can always let's do life. You are, uh, square square pass X square. Remember, the X is a constant. Okay, so you have are divided by square class X square. Okay. Yeah, the, uh minus, you know, to Ara. Okay, So that tend to be differentiated would be differentiated in. Yeah, the second part eso this hour will be gone. Okay. And then, uh, yeah, integrating this so you have to square root square past X square in our uh huh. So can you myself bring the two out. Okay, then, uh, zero to argue. Limits. Okay. Okay. So the first time we can start to put in the limits. So we have our on square root our square big r squared X square, And then the zero term is gone because of the small are here he and then turn behind. You have zero to our square are square plastic X square. They are okay as we continue. Okay, so the term in the back that we need to integrate, um, you can check the back of the book. Uh, this off integration table. There's a integration table, and so you can actually find answer. Uh huh. So this is half times, uh, are square are square plus x square. Yeah. And class X square, lawn R plus our square by square plus x square. Then there is not again. There's a swear again from zero to are there Closing bracket King two K e c pie are square with the square X square. Like this. Okay, sort of first time here Can we put in the limits? Are the gag. There are times, half times We are times square off our square plus x square Here again the zero term is gone because of the small are in front He then we continue to do that. Tell me the back, but we have half x square at your log. All right, are us square Roots are square plus x square. He us Yeah. Have x square. Ah, natural law. Uh, okay. Remember in this question access a constant. Okay. Okay. To E c high. Uh huh. Half our school roots r squared plus x square minus a half X quay. Um, that's a lot R plus square roots R squared plus x square. Bye bye. Eggs. Uh huh. And you can take out the house inside the bracket. He Then you get the finance. Er Okay, so this is, uh right. Answer me disk. That's how we arrive at the answer. Okay. And that's all

Okay, so does Chapter 22 problems 49 here and this says car just distributed within a solid sphere of radius are not in such a way that the charge density is a function of the radio position. So it's given as the charged density as a function of our is ro, not times are over, arm out. So if the total charge within this sphere is Q and his positive, what is the electric field everywhere within the sphere in terms of Q, are not an r Okay, So first we know that the symmetry of the charge distribution allow was this to be calculated using a gal seen spherical surface. So for a radius of are less than are not enclosed, charge is going to be the inner girl of our charged density function Times TV. So we can rewrite this in terms of just the radio distance, because we know that the volume and row on depend on the radio distance. So the smell and a greeting from zero to our says row, not times are prime over are not, and Devi, when we're changing into the are we could take up to four pi r squared. That's our problem. Squared and the are prime. So it becomes our inner grown out to solve for what the enclosed charges and we can actually just solve for this. So this becomes Piemonte over are not times four pi in a girlfriend zero to our of our prime cube yard prime. If we integrate that we get ro not pi r to the fourth over are not so this is our charge enclosed as a function of our So now we can right this in terms of the total charge So we know that q equals cute and closed at our equals are not so that is row, not pie are not cute. So if we want to write charge enclosed as a function of our this now becomes the total charge as the function better times are over are not all to the fourth Cool. So now we have the charging closed as a function of our in a queue. So let's add this new page here Nogales Ian's gases Laws Given his deed. Since we have a spherical symmetrical problem, we can use a spherical galaxy in surface and say that this is equal to e times the or pi R squared The surface area of our spherical surface that equals cute enclosed is a function of our over absolute, not SAR E now becomes Q over for pie. Just let him die. One over R Squared Times are over are not to the fourth or weaken right. It's better you over for pie. Absolutely not. R squared over R not to the fourth, and this is E as a function of our we see that it's positive. So we know that the electric field is pointing Radio Lee outward cool.


Similar Solved Questions

5 answers
7.1 Let W = Span{V1, V2, Explain why W € /and H be any vector space containing elements {V1,Vz,Vp} _
7.1 Let W = Span{V1, V2, Explain why W € / and H be any vector space containing elements {V1,Vz, Vp} _...
5 answers
You need to dilute your concentrated curcumin solution to get a new absorbance measurement: You have a 1 mL sample and you decide to dilute your solution by a factor of 10. How many milliliters of solvent do you need to add to your curcumin sample? (Hint: Final Volume Solute Volume = Dilution Factor)Select one: a. 5 mLb. 9 mLc10 mLd: 2 mLe: 20 mL
You need to dilute your concentrated curcumin solution to get a new absorbance measurement: You have a 1 mL sample and you decide to dilute your solution by a factor of 10. How many milliliters of solvent do you need to add to your curcumin sample? (Hint: Final Volume Solute Volume = Dilution Factor...
5 answers
Wpj Suppose XY have joint density functionC0 <* < 0 andy > 0e~r/2-7y fkxy) = 0else.(a) Are X and Y independent? (b) Find the probability P(X > 2,Y < 1).
Wpj Suppose XY have joint density function C0 <* < 0 andy > 0 e~r/2-7y fkxy) = 0 else. (a) Are X and Y independent? (b) Find the probability P(X > 2,Y < 1)....
5 answers
J1if < e 1A(z) 0 if z &
J1if < e 1A(z) 0 if z &...
5 answers
What Is the pH 0f0.41 M acelic acdsoJium €-eaic NaCH, COz nas been adcod? (K fcr acetic ac4 Is 1.8 * 107
What Is the pH 0f0.41 M acelic acd soJium €-eaic NaCH, COz nas been adcod? (K fcr acetic ac4 Is 1.8 * 107...
5 answers
Consider the following topic of interest. Identify the "failure".Are you right-handed left-handed, or ambidextrous? Research shows that approximately 10% ofthe world $ population is left-handed.Being right-handedNot bein richt-handedBcing ambidextrousBcing Icft-handedNot belg ambidextrousNot bcing Velt-handed
Consider the following topic of interest. Identify the "failure". Are you right-handed left-handed, or ambidextrous? Research shows that approximately 10% ofthe world $ population is left-handed. Being right-handed Not bein richt-handed Bcing ambidextrous Bcing Icft-handed Not belg ambidex...
5 answers
1J) (10 points) solution prepared bv What is the dhasolting freezing point of the 48,00 % of glycerin (C3HsO3) solution 210 & of ethunol (CzHsOH) mdni (reezing ~point-depression The freezing [int of pure Othanol constant (Kf) for ethanol 114.6 "Cat | alm Tlu: ctnunolare 92.[ g/mol and 46.1 g/mol, 1,99 Ch: The molar massts of glycerin and respectively.
1J) (10 points) solution prepared bv What is the dhasolting freezing point of the 48,00 % of glycerin (C3HsO3) solution 210 & of ethunol (CzHsOH) mdni (reezing ~point-depression The freezing [int of pure Othanol constant (Kf) for ethanol 114.6 "Cat | alm Tlu: ctnunolare 92.[ g/mol and 46....
5 answers
There areDe Morgan S Iaws 7 (A ^ 0) i5 eqvivalent to 7 Av 7B 2 7(A V B ) iS equivalent + 7A ^ 7 B shov tha+ We can derive Hc 2"4 Denorgan's cule trom t first on e
There are De Morgan S Iaws 7 (A ^ 0) i5 eqvivalent to 7 Av 7B 2 7(A V B ) iS equivalent + 7A ^ 7 B shov tha+ We can derive Hc 2"4 Denorgan's cule trom t first on e...
5 answers
Obtai tne 6.s .Xcos y dx + tagdy=0Obtexq Te 6,$ (ttLnx) dx + (1+by] dy =6
Obtai tne 6.s . Xcos y dx + tagdy=0 Obtexq Te 6,$ (ttLnx) dx + (1+by] dy =6...
5 answers
Contaminated water is being pumped continuously Linto a tankat a rate that Is Inversely proportional to the amount of water In the tank; that Is dv where y Is the number of gallons of water inthe tank after t minutes (t 2 0) Initially, there were 5 gallons of water In the tank and after 3 minules there were 7 gallons. How many gallons of water were in the tank at t = 18 minutes?V97V61
Contaminated water is being pumped continuously Linto a tankat a rate that Is Inversely proportional to the amount of water In the tank; that Is dv where y Is the number of gallons of water inthe tank after t minutes (t 2 0) Initially, there were 5 gallons of water In the tank and after 3 minules t...
5 answers
Write the first line of the declaration for a poodle class. The class should be derived from the Dog class with public base class access.
Write the first line of the declaration for a poodle class. The class should be derived from the Dog class with public base class access....
6 answers
The types of intermolecular forces in a substance are identical whether it is a solid, a liquid, or a gas. Why then does a substance change phase from a gas to a liquid or to a solid?
The types of intermolecular forces in a substance are identical whether it is a solid, a liquid, or a gas. Why then does a substance change phase from a gas to a liquid or to a solid?...
5 answers
Hcw long would il take to deposit 9,.94 & of copper (molar Fa 63.55 glmol) Jt the cathode Of an electrolytic cell; If a current ot 170 Ais through an aqucouS solution 0t CuSO4'321 min363 min290 min362,
Hcw long would il take to deposit 9,.94 & of copper (molar Fa 63.55 glmol) Jt the cathode Of an electrolytic cell; If a current ot 170 Ais through an aqucouS solution 0t CuSO4' 321 min 363 min 290 min 362,...
4 answers
3Question 5_ (8+12+8pts.) Let A =~2 9 - 8- 5_a. Find the eigenvalues of the matrix A. Show your work.b. Find basis for each of the eigenspaces of the matrix A. Show your work:Determine whether or not the matrix A is diagonalizable If the matrix A is diagonal- izable, find a nonsingular matrix P and diagonal matrix D such that D = P-IAP .
3 Question 5_ (8+12+8pts.) Let A = ~2 9 - 8- 5_ a. Find the eigenvalues of the matrix A. Show your work. b. Find basis for each of the eigenspaces of the matrix A. Show your work: Determine whether or not the matrix A is diagonalizable If the matrix A is diagonal- izable, find a nonsingular matrix P...
5 answers
2_ For the function g(x) e2xa) Find g' (x), g" (x),g" (x), g()(x) 6) Find the fourth degree Taylor_Polynomial centered about x = c) Find the Taylor_Series centered about x 0. Make sure to express the general term of the series and give the range of values for the index. You can also write your answer in Sigma notation such as problem #L_
2_ For the function g(x) e2x a) Find g' (x), g" (x),g" (x), g()(x) 6) Find the fourth degree Taylor_Polynomial centered about x = c) Find the Taylor_Series centered about x 0. Make sure to express the general term of the series and give the range of values for the index. You can also ...
5 answers
SaHoudredu; rOwcd his arc Juc Iromn hlama thc hurju crab trps Golng Cown thc bJyo; ne cauah: decreascd hls normal spced fallng (Ide that Incrcased hls ncrnia Going with tle tdc, the tnp took Minuicablt Going eJunst the tlce Tnnhont nunve-but coiriraunatFjadsthc plroquemph (Roundthc ncarcst hundrcath )How =Itom Boudreaux? Cimdcrab (ropalmilc; (Ruunotha nedre entheActmm
Sa Houdredu; rOwcd his arc Juc Iromn hlama thc hurju crab trps Golng Cown thc bJyo; ne cauah: decreascd hls normal spced fallng (Ide that Incrcased hls ncrnia Going with tle tdc, the tnp took Minuicablt Going eJunst the tlce Tnnhont nunve- but coirira unat Fjads thc plroque mph (Round thc ncarcst hu...
5 answers
Use the Root Test to determine il the following series converges absolutely Or divergesic-(.% (Hinl lim (1+x/m)" e* ) n-0 Sinca Ihe Iimit resulting from Ihe Rool Test is (Type an exact answor )
Use the Root Test to determine il the following series converges absolutely Or diverges ic-(.% (Hinl lim (1+x/m)" e* ) n-0 Sinca Ihe Iimit resulting from Ihe Rool Test is (Type an exact answor )...

-- 0.027099--