Electric Field Notes PDF

ELECTRIC FIELD Electric Field Electric force is a noncontact force. An electric charge q can exert force on other charged objects even though they are at some distance away. The space surrounding a charged body is called an electric field. An electric field causes any charged par...

ELECTRIC FIELD Electric Field Electric force is a noncontact force. An electric charge q can exert force on other charged objects even though they are at some distance away. The space surrounding a charged body is called an electric field. An electric field causes any charged particle placed in it to experience an electric force. Every charge has an electric field associated with it. Electric Lines Force Michael Faraday, an English scientist who made important discoveries in electricity and magnetism, introduced the use of electric lines of force to map out an electric field. Lines of force have the following properties: Electric Lines Force Lines of force have the following properties: 1. Lines of force start from positively charged particles and end on negatively charged particles or continue toward infinity. 2. Lines of force neither intersect nor break as they pass from one charge to another. Electric Lines Force Lines of force have the following properties: 3. The greater the number of lines of force, the stronger the electric field. The neutral point is the point where no lines of force pass. The electric field is zero at the neutral point. Thus, neutral points are points where the resultant field is subtractive and the electric fields are equal but oppositely directed. A neutral point between two like charges is a point between the two charges and nearer the smaller charge. For two unlike charges, lines of force can pass from positive to negative charge. The neutral point cannot be between them; it is an external point along the line joining them and nearer the small charge. Electric Lines Force Lines of force have the following properties: Electric Field Due to a Point Charge An electric field exists in the region of The magnitude of the space around a charged object or a electric field due to the source charge. When another point charge is: charged object enters this electric field, it will experience an electric force. The strength of the electric field 𝑞 at a point charge is called electric 𝐸=𝑘 ! field intensity. 𝑟 Electric Field Due to a Point Charge E is the magnitude of the electric field 𝑞 in N/C. 𝐸=𝑘 ! k coulomb’s constant 𝑟 𝒒 magnitude of the point charge in C r is the distance from the charge to the point where the field is measured in m. Sample Problem 1 Find the electric field 2 meters away from a charge of +5 𝜇𝐶  Sample Problem 2 Calculate the magnitude and direction of the electric field 0.45 m from a + 7.85 𝑥 10!" 𝐶 point charge.  Sample Problem 3 An electric dipole consists of two equal but unlike charges separated by a distance. Two point charge 𝑞# = +4.5𝑥10!$ 𝐶 and 𝑞% = −4.5𝑥10!$ 𝐶, are separated by 6.4 𝑥 10% 𝑚 , forming an electric dipole as shown in the figure. Find the electric field halfway between the dipole. 6.4 𝑥 10" 𝑚 𝑟! 𝑟" + -  AWARE OF OUR POTENTIALS Electric potential and capacitance are closely related to each other. Learning Objectives: Define electric potential energy, electric potential, and capacitance Compute the electric potential created by point charges and continuous charge distribution Compute the capacitance of a parallel plate capacitor Compute the equivalent capacitance of capacitors when connected in series, parallel or combination of series and parallel Electric Potential U is the electric potential energy (in joules, J) The electric potential (also called electrostatic V is the electric potential at point (in potential or potential) at any point in electric volts, V) field E is electric potential energy per unit charge at that point. Electric potential is 𝑈 designated as V. 𝑉= Electric potential is a scalar quantity. Its SI unit 𝑞! is the volt (V) named after Italian physicist Alessandro Volta who devised one of the first electric cells. Note that 1 V is equal to 1 q is the charge (in joule/coulomb (J/C) coulombs, C) Electric Potential due to a Point Charge: For a point charge q, the electric potential V at a distance r from the charge is given by: 𝑘𝑞 Where: 𝑉= 𝑘 𝑖𝑠 𝐶𝑜𝑢𝑙𝑜𝑚𝑏’𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (9.00 𝑥 10* 𝑁. 𝑚+/𝐶 +) 𝑟 𝑞 𝑖𝑠 𝑡ℎ𝑒 𝑐ℎ𝑎𝑟𝑔𝑒 𝑐𝑟𝑒𝑎𝑡𝑖𝑛𝑔 𝑡ℎ𝑒 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑓𝑖𝑒𝑙𝑑 𝑟 𝑖𝑠 𝑡ℎ𝑒 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑐ℎ𝑎𝑟𝑔𝑒 Potential Difference (∆𝑉)(𝑉𝑜𝑙𝑎𝑡𝑔𝑒) Potential difference (often referred to simply as voltage) is the difference in electric potential between two points in an electric field. It represents how much work is needed to move a charge between those two points. Voltage is a measure of the energy per unit charge required to move a charge from one point to another. Scalar Quantity: Like electric potential, potential difference is a scalar. Potential Difference (∆𝑉)(𝑉𝑜𝑙𝑎𝑡𝑔𝑒) Units: The unit of potential difference is also the volt (V), and it is measured in the same way as electric potential: 1 volt = 1 joule per coulomb. Work and Energy: The potential difference between two points indicates how much energy is required to move a unit positive charge between those two points. If a charge moves in the direction of decreasing potential, the system does positive work on the charge. Relation to Electric Field: The potential difference between two points is related to the electric field along the path connecting those points. Potential Difference (∆𝑉)(𝑉𝑜𝑙𝑎𝑡𝑔𝑒) Formula for Potential Difference: The potential difference between two points, A and B, in an electric field is: ∆𝑽 = 𝑽𝑩 − 𝑽𝑨 Where: - ∆𝑉 is the potential difference (voltage), - 𝑉, 𝑎𝑛𝑑 𝑉- are the electric potentials at point B and A, resepectively. Relationship Between Electric Potential and Electric Field: The electric field E is related to the potential difference ∆𝑉 and the distance d between two points: ∆𝑉 𝐸=− 𝑑 Where: - E is the electric field (in volts per meter, V/m) - ∆𝑉 is the potential difference between the points (in volts, V) - d is the distance between the two points (in meters, m). This formula shows that the electric field is the negative gradient of the potential. A positive field points from higher to lower potential.  Sample Problem 1 Consider a point charge q = 2 𝜇𝐶 and calculate the electric potential and potential difference at two points: one at a distance of 1 meter and another at 3 meters from the charge.  Sample Problem 2 A point charge of −10 𝑥 10!" 𝐶 is 5.00 m from point A and 8.00 m from point B. (a) Find the potential at point A and B. (b) How much work is done by the electric field in moving a 3.00 nC particle from point A and B?  Sample Problem 3 A point charge of 𝑞 = 5𝜇𝐶 is placed at the origin. What is the electric potential at a point 2 meters away from the charge? CAPACITORS  Capacitors One important element in an electric circuit is a capacitor. A capacitor is a device for storing charges. The standard symbols for a capacitor are shown in figure. There are several types of capacitors. One of the simplest types of capacitors consists of two equally but oppositely charged parallel conducting plates separated from each other by a thin sheet of insulating material or dielectric.  Capacitors When connected to a source of charge, such as a battery, the positive terminal of the source removes electrons from the plate connected to it and transfers them to the other plate. As a result, the two plates are equally but oppositely charged. Figure shows the basic parts of a parallel plate capacitor. A capacitor is usually named after the dielectric material used. Common dielectric materials used in a capacitor are mica, glass, air, ceramic, and paper.  Capacitance Capacitance is the ability of a capacitor to store charges. The capacitance C of a capacitors is mathematically defined as the ratio of the amount of charge q in one plate to the potential difference V between the plates. In symbols, Where: 𝑄 𝐶 = 𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑎𝑛𝑐𝑒 (𝑖𝑛 𝑭𝒂𝒓𝒂𝒅𝒔, 𝐹) 𝐶= 𝑄 = 𝐶ℎ𝑎𝑟𝑔𝑒 𝑠𝑡𝑜𝑟𝑒𝑑 (𝑖𝑛 𝑪𝒐𝒖𝒍𝒐𝒎𝒃𝒔, 𝐶) 𝑉 = 𝑉𝑜𝑙𝑡𝑎𝑔𝑒 𝑎𝑐𝑟𝑜𝑠𝑠 𝑡ℎ𝑒 𝑝𝑙𝑎𝑡𝑒𝑠 (𝑖𝑛 𝑽𝒐𝒍𝒕𝒔, 𝑉) 𝑉 The SI unit of capacitance is the farad (F) named after Michael Faraday. Note that 1 farad is equal to 1 coulomb per volt.  The capacitance of parallel plate capacitor is affected by the following factors: a. The area of plates. The bigger the area of the plates, the greater the capacitance. b. The distance between the plates. The closer the plates to each other, the greater the capacitance. c. The insulating material or dielectric between them. The capacitance is determined in terms of the material's permittivity constant 𝜖 the higher the 𝜖 , the greater the capacitance. The dependence of the capacitance of a parallel plate capacitor on the factors cited is mathematically expressed as: 𝐴 𝐶 =∈ 𝑑 - C = capacitance in Farads, F 1 - ∈.= 𝑃𝑒𝑟𝑚𝑖𝑡𝑡𝑖𝑣𝑖𝑡𝑦 𝑜𝑓 𝑓𝑟𝑒𝑒 𝑠𝑝𝑎𝑐𝑒 8.854 𝑥 10/0+ 2 - ∈3 = relative permittivity (dielectric constant) of the material between the plates - A = area of one plate in 𝑚+ - d = separation between the plates in m  𝐴 𝐶 =∈ 𝑑 Dielectric 𝝐(𝒙𝟏𝟎#𝟏𝟏 𝑪𝟐 / 𝑵. 𝒎𝟐 ) Air or vacuum 0.885 Glass (ordinary) 7 Glass (pyrex) 4.7 Mica 4.8 Mylar 2.7 Paraffin 2 Polyethylene 1.99 Porcelain 6.2 Teflon 1.9 Water at 20! 𝐶 70.8  Sample Problem 1 A capacitor consists of two square metal pates, each measuring 5.00𝑥10!% 𝑚 on a side. In between the plates is a sheet of mica measuring 1.00 𝑥 10!N 𝑚 thick. (a) What is the capacitance of this capacitor? If ten charge in one plate is 2.00 𝑥 10!O 𝐶, what is the (b) potential difference and (c) electric field between the plates?  Sample Problem 2 (a) What is the capacitance of a parallel plate capacitor with metal plates, each measuring 1.00𝑚% , separated by 1.00 mm? (b) What charge is stored in this capacitor if a voltage of 3.00𝑥10P V is applied to it? Combination of Capacitors Capacitors may be connected in series and parallel. Referring to figure, the series combination of capacitors is characterized by only one path for charge transfer through terminals A and B. All the series capacitors acquire the same charge. The charges in each capacitor are equivalent, and are all equal to the total charge in the combination. But because they have different capacitances, the potential differences between the plates of the capacitor are different  In summary, the following relationship apply for capacitors in series a. Charge: 𝑞QRQST = 𝑞# = 𝑞% = 𝑞P = ⋯ 𝑞U b. Potential difference: 𝑉QRQST = 𝑉# + 𝑉% + 𝑉P +.. 𝑉U V c. Capacitance: Using the equation 𝐶 = and the above relationship between W &!"!#$ &% && &' &( charges and voltages, = + + +.. + '!"!#$ '% '& '' '( 1 1 1 1 1 = + +..+ 𝐶QRQST 𝐶# 𝐶% 𝐶P 𝐶U Capacitors may be connected in series and parallel. For parallel capacitors, there are several paths for the transfer of charges through the voltage terminals A and B. since the capacitors are connected to the same terminals A and B, the potential differences between their plates are equivalent, and are equal to 𝑉QRQST In summary, the following relationship apply for capacitors in parallel a. Charge: 𝑞QRQST = 𝑞# + 𝑞% + 𝑞P + ⋯ 𝑞U b. Potential difference: 𝑉QRQST = 𝑉# = 𝑉% = 𝑉P =.. 𝑉U V c. Capacitance: Using the equation 𝐶 = and the above relationship between W charges and voltages, 𝐶QRQST 𝑉QRQST = 𝐶# 𝑉# + 𝐶% 𝑉% + 𝐶P 𝑉P +.. +𝐶U 𝐶QRQST = 𝐶# + 𝐶% + 𝐶P +.. +𝐶U Sample Problem 1 Given that 𝐶( = 10.0 𝐹, 𝐶) = 5.0 𝐹, 𝑎𝑛𝑑 𝐶* = 4.0 𝐹, find the total capacitance for each connection shown. 𝑪𝟏 𝑪𝟐 𝑪𝟑 Sample Problem 2 Given that 𝐶( = 10.0 𝐹, 𝐶) = 5.0 𝐹, 𝑎𝑛𝑑 𝐶* = 4.0 𝐹, find the total capacitance for each connection shown. 𝑪𝟏 𝑪𝟐 𝑪𝟑 Sample Problem 3 Given that 𝐶( = 10.0 𝐹, 𝐶) = 5.0 𝐹, 𝑎𝑛𝑑 𝐶* = 4.0 𝐹, find the total capacitance for each connection shown. 𝑪𝟏 𝑪𝟑 𝑪𝟐 Sample Problem 4 Two capacitors with 2.0 F and 3.0 F capacitance, respectively, are connected in series and subjected to a total potential difference of 100 V. Find the (a) total capacitance, (b) charge stored in each capacitor, and (c) potential difference across each capacitor. Sample Problem 5 Two capacitors with 5.0 F and 4.0 F capacitance, respectively, are connected in parallel instead of in series. The combination is connected to a 100 V line. Find the (a) total capacitance, (b) charge stored in each capacitor, and (c) potential difference across each capacitor.