AP Physics - Coulomb's Law

Transcription

1 AP Physics - oulomb's Law We ve leaned that electons have a minus one chage and potons have a positive one chage. This plus and minus one business doesn t wok vey well when we go in and ty to do the old majo figuing stuff out deal it s kind of abitay thing, see. So we need a eally useful unit of chage. Well, we got one. It is you basic standad unit of chage; a thing called a oulomb (). The symbol fo chage is Q howeve is used as well. One oulomb is eual to the chage of 6.5 x 8 electons o potons. The chage of a single electon is -.6 x -9. The chage of a poton is +.6 x -9. The oulomb is a lage amount of chage, so it is vey common to use mico oulombs and milli olulombs. m = -3 = -6 What is the chage of.35 x 7 electons? This is a simple dimensional analysis type poblem x.35 x e.6 x e o.6 x.6 x.6 3 x oulomb s Law: haged objects exet foces on one anothe. This is vey simila to what happens with gavity between two objects that have mass. Recall that ewton s univesal law of gavity can be used to calculate the foce between two objects that have mass. It tuns out that thee is a simila law that can be used to calculate the foce between two objects that have chage. This law is called oulomb s law. Hee it is: 4 is the foce exeted between the two chages. and ae the two chages. (ote, we will actually use the absolute value of the chages - we simply don t cae about whethe they ae positive o negative.) is the distance between the two 84

2 chages and is called oulomb s onstant. It is simila to the univesal 4 gavitational constant. The value fo oulomb s constant is: 8.99 x 4 9 m oulomb s law in most physics books is usually witten in a slightly diffeent fom: ke k o e whee k e x 9 m But we won t use that fom, because the wondeful AP Physics folks use the fist one. The foce between two chaged objects can be eithe attactive o epulsive, depending on whethe the chages ae like o unlike. We will also assume that the chages ae concentated into a small aea point chages. Example Poblems: Two point chages ae 5. m apat. If the chages ae. and.3, what is the foce between them and is it attactive o epulsive? m 9 m 8.99 x x. x The foce is epulsive - both chages ae positive. A foce of.6 x -3 exists between chages;.3 and 3.5. How fa apat ae they? 85

3 4 4 9 m x.3 x 3.5 x x 5.57 x m 5. m Electic oce and Gavity: Both gavity and the electic foce ae fundamental foces. The euations fo the gavity and the electomagnetic foce have the same fom; they ae both invese suae elationships. 4 and Whee the gavity. 4 Gmm tem is the constant fo oulomb s law and G is the constant fo the law of Thee ae fou eally significant diffeences in the two foces: Gavity is always attactive. The electomagnetic foce can be eithe attactive o epulsive. Gavity is much weake. Gavity has a much geate ange within which it is a significant foce. The electic foce can be shielded, blocked, o cancelled. You cannot do any of these things with the gavity foce. The foce of gavity is aound 4 times smalle than the electomagnetic foce. This can be seen in a compaison of the two popotionality constants. 4 = 8.99 x 9 m / fo the electic foce 86

4 G = 6.67 x - m /kg fo the gavitational foce The two constants diffe by a facto of! Gavity extends out to geat distances. The sun is 9 million miles away fom the eath, yet the foce of gavity is lage enough to cause the eath to be locked in an obit aound the sun. ow fo electicity, the foce between two chages dops off vey uickly with distance. This is because the magnitude of the two chages is vey small at the most, maybe a few oulombs. But with gavity, we ae dealing with enomous masses - the mass of the sun is.99 x 3 kg! Because of these lage masses, even though the gavity constant is vey small, the foce of gavity between eally lage masses ends up being a eally big foce that eaches out ove distances of billions and even tillions of kilometes. Let us compae foces in a hydogen atom. The hydogen atom is made up of a poton and an electon. The two paticles attact each othe because they both have mass and they also have opposite chages. The magnitude of the electon/poton chage is.6 x -9. The distance between them in a hydogen atom is aound 5.3 x - m. o the mass of an electon we ll use 9. x 3 kg. o the mass of a poton we ll use.7 x 7 kg. We can now calculate the foce of gavity between the two paticles: x kg 9. x kg 5.3 Gm m kg x m m 6.67 x x (Petty small, ain t it?) ext, we solve fo the electomagnetic foce x.6 x m 8.99 x x m x 8. x Looking at the two foces, we see that gavity is much weake. The electomagnetic foce is. x 39 times bigge! Gavity vs Electicmagnetic oce: Gavity fields and electic fields ae both simila and diffeent at the same time. Hee is a handy little table to oganize things: 87

5 Gavity oce Attacts invese suae law suound objects cannot be shielded incedibly weake Electicmagnetic oce attacts and epels invese suae law suound objects can be shielded enomously stonge Supeposition Pinciple: When we have moe than two chages in poximity, the foces between them get moe complicated. But, please to elax, even though things seem complicated, they actually ain t and it is faily simple to wok things out. The foces, being vectos, just have to be added up. We call this the supeposition pinciple. Supeposition Pinciple The esultant foce on a chage is the vecto sum of the foces exeted on it by othe chages. Let s look at a system of thee chages. The chages ae aanged as shown in the dawing. is 3. m fom. is 4. m fom the 3. (We immediately spot this as one of those 345 tiangle deals, so we know that is 5. m fom 3 ). What is the net foce acting on 3? 3 4. m m + 5. m 3 is attacted to (they have opposite chages) and epulsed by (they have the same chage). The two foce vectos have been dawn and labeled, 3 and x 9. x x The net foce on 3 is The fist step is solving the poblem is to find the magnitude of the foces: x 6. x 9 m x 4 5. m x.8 x 88

6 9 9. x 5. x 9 m x 4 4. m x The next step is to beak the two vectos down into thei hoizontal and vetical components and add the two vectos in the x and y diections. This gives us the components of the esultant vecto, X and Y : cos x x cos x x x 5.6 x 3. x x y 3 sin 8 o 9.8 x sin x y ow we can find the esultant foce: y x x x x sin cos x ow we can find the diection o the esultant foce: Y tan X x tan 9 3. x 65. with the x axis Isn t it geat to solve these poblems? Let s do anothe! Two.5 g metal sphees have identical positive chages. They hang down fom light stings that ae 3 cm long commonly attached to the ceiling. If the angle the stings fom with the vetical is 7., what is the magnitude of the chages? 3 cm 89

7 T ist let s daw a BD fo the foces on one of the sphees: mg E Thee ae thee foces: the tension T in the sting holding the sphee up, the electic foce E, and the weight of the sphee mg. Because the system is static, the sum of the foces must be zeo. The foces in the x diection ae: Tsin E The foces in the y diection ae: Tcos mg ow things become simple, we can use this last euation to find the tension. Amed with the tension we can find the electic foce. Using the electic foce we can find the chage on the sphee. Let s go ahead and find the tension. mg m Tcos mg T.5 kg o cos s cos 7. ow we find E : o Tsin Tsin.47 sin 7..3 E oulomb s law is next: E 4 The chages have the same value; we need to find the distance between the two sphees. l l=3 cm We can see that half of is l sin. So is l sin. 4 same, so: But the two chages ae the 9

8 4 4 4 sin l 4 4 o.3 m sin x 9 m x x.43 x 7.43 x 9