Electrostatics (Electric Charges and Field) #2 2010

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1 Electic Field: The concept of electic field explains the action at a distance foce between two chaged paticles. Evey chage poduces a field aound it so that any othe chaged paticle expeiences a foce when kept in the field. An electic field is said to exist at a point if an electic foce is expeienced by a test chage (any chaged body) kept at that point. Quantitatively, the electic field at a point is defined as the foce expeienced pe unit chage by a positive test chage kept at that point. Mathematically, Electic Field is a vecto quantity and its diection is same as the diection of foce expeienced by a positive test chage kept at that point. The unit of electic field is newton pe coulomb (N C -1 ) o volt pe mete (V m -1 ) The dimension of electic field The foce expeienced by a chage q in a field F=qE A positive chage will expeience a foce in the diection of the field and a negative chage will expeience a foce opposite to the diection of the field. Electic field due to a point chage. Conside a point chage q kept at a point. Let be a point at a distance fom the point chage whee we wish to detemine the electic field. If a test chage q 0 is kept at the point, the electic foce expeienced by q 0 is given by, Then, the foce expeienced by test chage due to dq, The total foce on q 0 Volume chage distibution If thee is a body with chage distibuted all ove its volume so that the chage pe unit volume (volume chage density) is, the chage on an elementay volume dv is dq= dv The foce on a test chage q 0 due to this elementay chage is, The total foce on q 0 by the chage distibution, The electic field at the location of q 0, Suface chage distibution If chage is distibuted in two dimensional space, then the suface density of chage If ds is a small elementay aea, then the chage on this elementay aea ds is dq= ds The foce expeienced by a test chage q 0 at a distance, The total foce on q 0 by the chage distibution, The electic field at the location of q 0, Theefoe, The electic field at, inciple of Supeposition of electic Field. The electic field at any point due to a numbe of point chages is equal to the vecto sum of electic fields poduced by each chage individually at that point. Electic Field due to continuous chage distibutions When a lage numbe of point chages ae kept vey close to each othe in a continuous manne, then a test chage kept at a distance will expeience a foce. If we imagine that the chage on the body is made up of lage numbe of elementay chages dq each, Linea Chage Distibution If chage is distibuted in one dimension, then we speak about linea chage density = dq/dl If we conside a small elementay length dl. the chage on it will be, dq= dl The foce on a test chage q 0, The total foce on q 0, The electic field at the location of q 0, w w w. p l u s t w o p h y s i c s. c o m age 1

2 Electic Dipole: An electic dipole is a pai of equal and opposite chages q and q sepaated by a small distance 2l Dipole moment () is the poduct of eithe chage and the distance between the two point chages. Electic dipole moment is a vecto quantity and is diected fom negative chage to positive chage. Unit of electic dipole moment is coulomb mete (C m) Electic Field due to a dipole (a) Axial osition Conside a dipole fomed by two chages ±q sepaated by a distance 2l. Let be a point on the axial line at a distance fom the cente of the dipole. The electic field at due to chage (-q) at A is towads left and is given by (along A) The electic field at due to chage (q) at B is towads ight and is given by (along B) So, the esultant electic field at will be towads ight since E B > E A E axial = E B - E A = Reaanging, 2lq =, the electic dipole moment of the dipole., Fo a shot dipole >>l and l 2 in denominato can be neglected in compaison to 2 moment (b) Equatoial osition A ( q) B (q) E A E B In the diection of dipole Conside a dipole fomed by two chages ±q sepaated by a distance 2l. Let C be a point on the equatoial line at a distance fom the cente of the dipole. The electic field at the point C due to (-q) at A is along CA (towads A) and the electic field at C due to (q) at B is along BC (away fom B). The two fields ae equal in magnitude but act in diffeent diections. A (-q) B (q) Resolving E A and E B in mutually pependicula components, the vetical components E A sin and E B sin cancel out being equal in magnitude and opposite in diection. The hoizontal components E A cos and E B cos add up. The esultant electic field on the equatoial position, But, E A Cos E B Cos E B E A E B Sin C E A Sin Fo a shot dipole >>l and l 2 in denominato can be neglected in compaison to 2 aallel to the dipole moment and opposite in diection (c) Any position (shot dipole) Conside an electic dipole of dipole moment. Let C be a point at a distance (>>l). The line joining the cente of the dipole and the point C make an angle with the axis of the dipole. C cos sin The dipole moment can be esolved in to two mutually pependicula components cos and sin. Fo the component cos C is axial position and fo the component sin C is equatoial position. Let the components of electic field at C be E ll and E w w w. p l u s t w o p h y s i c s. c o m age 2

3 Since they ae mutually pependicula, the esultant electic field at C, n axial position, =0, and cos =1 n equatoial position =90, and cos =0 Toque on a dipole in a unifom electic field Conside a dipole fomed by two chages ±q sepaated by a distance 2l placed in a unifom electic field E with its axis making an angle with q the electic field. qe B 2l cos qe -q The chage A q at B expeiences a foce qe in the diection of electic field and the chage q expeiences a foce qe opposite the diection of electic field. The foces ae equal in magnitude but opposite in diection. Theefoe the net foce acting on the dipole is zeo. But since these equal and opposite foces ae paallel to each othe sepaated by a distance, they constitute a couple. The toque (tuning effect) of the couple is given by In Vecto fom, The diection of is pependicula to both and E The toque ties to align the dipole moment paallel to the electic field. When =0, the dipole is in stable equilibium (net foce = 0 and net toque = 0 and the dipole egains its oiginal position when slightly distubed) and when = 180, the dipole in unstable equilibium (net foce = 0 and net toque = 0 and the dipole goes futhe away fom its oiginal position when slightly distubed) Toque is maximum when =90 ; Electic dipole in a non unifom field In a unifom electic field, the foce on the two chages of the dipole will be diffeent and theefoe the dipole will expeience a net foce as well as a net toque. Electic field lines Electic field lines ae used to epesent a field diagammatically. An electic field line (also called electic lines of foce) is an imaginay line in an electic field such that the tangent to it at any point gives the diection of electic field at that point. In an electic field a fee positive test chage will move along an electic field line. opeties of electic field line 1. They stat fom a positive chage and end in a negative chage 2. Field lines ae smooth continuous cuves without any abupt changes o beaks. 3. The tangent to it at any point gives the diection of electic field at that point. 4. Electic field lines neve intesect each othe* 5. They always ente o emege nomal to the suface of a conducto**. 6. They tend to contact longitudinally (on account of attaction between unlike chages) 7. They tend to expand lateally (on account of epulsion between like chages) 8. The elative closeness of field lines indicate the stength of electic field. i.e, Electic field lines ae close in egions of geate electic field and fa apat in egions of weake electic field. *Electic field lines neve intesect each othe. Why? If they intesect, two tangents can be dawn at the point of intesection indicating two diffeent diections of electic field at the same position which is impossible. **Electic field lines ae always nomal to the suface of a conducto. Why? If the electic field line is not nomal, it can be esolved into two components, a nomal component and a tangential component. The tangential component will make the fee chages of the conducto to move along the suface of the conducto. But no such movements ae obseved indicating that thee is no tangential component which implies that the electic field lines ae always nomal to the suface of a conducto. Aea vecto Aea vecto of a suface is a vecto nomal to the suface and whose magnitude is equal to the aea of the suface Solid angle The atio of the aea to the squae of the magnitude of adius vecto is the solid angle subtended by the aea. w w w. p l u s t w o p h y s i c s. c o m age 3

4 The solid angle subtended by an aea ds whose adius vecto makes and angle with the aea vecto is Fo an abitay suface, the total angle subtented by the suface The total solid angle aound a point in fee space is 4 std Electic Flux Electic flux though a suface is a measue of the total numbe of electic field lines passing nomal to the given suface. The electic flyx though and elementay suface of aea ds is given by The total electic flux though a suface Electic Flux is a scala quantity and its unit is Nm 2 C -1 Gauss Theoem Gauss theoem states that the total electic flux though a closed suface (in vacuum) is equal to times the net chage enclosed by the suface. Mathematically, Inside a medium the eqn becomes Applications of Gauss theoem (i) Electic field due to a point chage Conside a point chage q kept at the point. Let be a point at a distance fom the point chage. Imagine a Gaussian sphee** passing though with the chage at as the cente. Let ds be a small elementay aea on the suface of the sphee. q Applying Gauss theoem, By symmety, E is constant at any point on the suface of the Gaussian sphee and =0 since E and ds ae in the same diection., since Theefoe, The electic field at the point, (ii) Electic field due to a line of chage Conside a long staight line of chage of linea chage density. Let be a point at a distance fom the line of chage. Imagine a Gaussian Cylinde passing though and with the line of chage as the axis. Let ds be a small elementay suface aound the point. Applying Gauss theoem, The cylindical Gaussian suface is composed of thee sufaces two plane sufaces and one cuved suface. But =0 fo cuved suface and 90 fo plane suface. Theefoe, By symmety, E is constant thoughout the cuved suface and q = l, whee l is the length (height) of the Gaussian cylinde. Theefoe, E But, E R E =, cuved suface aea of the cylinde (iii) Electic field due to a plane sheet of chage Conside a plane sheet of chage of suface density of chage. Let be a point at a distance fom the plane sheet of chage. Imagine a Gaussian cylinde passing though and symmetic about the plane sheet of chage as shown in figue. w w w. p l u s t w o p h y s i c s. c o m age 4

5 Imagine small elementay aea ds aound. Applying Gauss Theoem, Theefoe, The electic field at the point The cylindical Gaussian suface is composed of thee sufaces two plane sufaces and one cuved suface. But =0 fo plane suface and 90 fo cuved suface. Theefoe, By symmety, E is constant thoughout the plane sufaces as we have imagined the two pane sufaces equidistant fom the sheet of chage and q = S, whee S is the aea of coss section of the cylinde and is the aea of the plane sheet inside the Gaussian cylinde. E is independent of distance (povided the distance is too lage compaed to the size of the sheet of chage) (iv) Electic field due to a chage spheical conducting shell Conside a chaged spheical conducting shell with chage q with cente. Hee, thee ae thee diffeent typical positions fo the point whee the electic field can be evaluated. 1. utside the spheical shell Let be a point outside the spheical shell at a distance fom the cente of the sphee. Imagine a Gaussian sphee passing though and concentic with the spheical shell. Let ds be a small elementay aea on the suface of the sphee. R q This value is same as that due to a point chage. Theefoe we can say that the chaged spheical shell behaves as if the entie chage is concentated at its cente. 2. n the spheical shell n the spheical shell, =R. Theefoe, 3. Inside the spheical shell Fo a point inside the sphee, the Gaussian suface does not enclose any chage and theefoe the Electic field = o Accoding to Gauss Theoem, Hee q enclosed =0 Theefoe, E = 0 This is applicable fo a cavity of any shape. The electic field inside a cavity is always zeo The above esults ae also tue fo a chaged solid conducting sphee as well as a unifomly chaged nonconducting spheical shell. Electostatic Shielding The vanishing of electic field inside a cavity is called electostatic shielding. The above esults ae also tue fo a chaged solid conducting sphee as well as a unifomly chaged non-conducting spheical shell. Gaph showing the vaiation of electic field with distance fom the cente of a chaged spheical shell E Applying Gauss theoem, By symmety, E is constant at any point on the suface of the Gaussian sphee and =0 since E and ds ae in the same diection., since R Q. Why is it safe to be inside the ca when you ae out in a ca duing the time of lightning and thunde? Q. Suggest a method to safely cay a highly sensitive electical instument though a egion of stong and vaying electic field. Explain the pinciple used in the method suggested by you. Q. Why the net chage inside a conducto is always zeo? **************************** w w w. p l u s t w o p h y s i c s. c o m age 5

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