Electric Field Summary

The stinky field If you have spent time around infants, you are likely familiar with the STINKY FIELD: Stronger effect when you’re closer Stronger effect with more concentrated source Stronger effect on better detectors

Stinky field  Electric field Stronger with more concentrated source Only repulsive Stronger closer to source More pronounced effect on more sensitive detectors Detected with test charges Comes in discrete units Detected with noses or gas chromatographs Can be attractive

Electric Field 𝐸 ≝ 𝐹 𝑞 The strength of the electric field 𝐸 is defined as the force exerted on a miniscule test charge ( 𝑙𝑖𝑚 𝑞→0 ) . The electric field strength describes the effect of the charges creating the electric field (and we ignore the electric field created by the test charge) Reported in units of Newtons per Coulomb (N/C)

Quantifying Electric Field 𝐸 = 𝐹 𝑞 where 𝐹 =𝑘 𝑄𝑞 𝑟2 So, 𝐸 = 𝑘𝑄𝑞 𝑟2 𝑞 𝐸 =𝑘 𝑄 𝑟2 Strength of electric field is directly proportional to strength of source charge Q Strength of electric field is inversely proportional to the square of the distance from the source charge

Example A proton (q=1.6 x 10-19 C) is released in a uniform electric field and experiences an electric force of 1.86 x 10-14 N toward the south. What is the magnitude of the field? What is its direction? 𝑞=1.6 𝑥 10 −19 𝐶 𝐹 = 1.86 𝑥 10 −14 𝑁 𝐸 =? 𝐸 = 𝐹 𝑞 = 1.86 𝑥 10 −14 𝑁 1.6 𝑥 10 −19 𝐶 𝐸 =1.2 𝑥 10 5 𝑁 𝐶 south

Example Determine the magnitude of acceleration experienced by an electron (q=-1.6 x 10-19 C, m = 9.1 x 10-31 kg) in an electric field of 756 N/C. 𝑞=−1.6 𝑥 10 −19 𝐶 𝑚=9.1 𝑥 10 −31 𝑘𝑔 𝐸 =756 𝑁 𝐶 𝑎=? 𝐸 = 𝐹 𝑞 where 𝐹 = 𝑚𝑎 So, 𝐸 = 𝑚𝑎 𝑞 so, 𝑎 =𝐸 𝑞 𝑚 𝑎 =(756 𝑁 𝐶 ) −1.6 𝑥 10 −19 𝐶 9.1 𝑥 10 −31 𝑘𝑔 𝑎 = 1.32 𝑥 10 14 𝑚 𝑠 /𝑠

Electric Potential

Difference in Electrical Potential Energy In the case of a spring or an electric charge, ∆𝑈=−𝑊 where 𝑊= 𝐹𝑑 So, ∆𝑈 =−𝐹𝑑 where 𝐹=𝑞𝐸 So, ∆𝑈=−𝑞𝐸𝑑

Difference in Electrical Potential Energy In the case of an electric charge, ∆𝑈=−𝑞𝐸𝑑 Change in potential depends on the distance between the two charges. Change in potential depends on the strength of the electric field Change in potential depends on size of test charge

Difference in Potential Energy / Charge When describing forces induced by electric fields, it was useful to introduce 𝐸 as the strength of field in terms of force per charge, 𝐹 𝑞 . When dealing with differences in potential energy, it is useful to describe measure it in terms of energy difference per charge.

Difference in Potential Energy / Charge We introduce a new term for difference in potential energy per charge: voltage, V V≝ ∆𝑈 𝑞 Measured in Joules per Coulomb Abbreviated as volts, V Frequently referred to as ‘potential difference’ Named in honor of Alessandro Volta (1745 – 1827), inventor of first batteries

Quantifying voltage – part 1 V= ∆𝑈 𝑞 So, ∆𝑈=V𝑞 How much work can a given charge do?

Cliff  Potential difference height = potential energy mass = potential energy ∆𝑈 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 = 𝑚𝑔ℎ voltage= potential energy charge= potential energy ∆𝑈 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 =𝑞V

Typical voltages 108 V 105 V 1.5 to 9 V 10-4 V 120 V 106 V Thundercloud to ground 108 V High-voltage powerline 105 V batteries 1.5 to 9 V Potential changes on skin (EKG) 10-4 V Household voltage (US) 120 V Van der Graaf generator 106 V

Example A electron (q=1.6 x 10-19 C) is accelerated through a potential difference of +5000 V in old television. What is the change in electrical potential energy? 𝑞=−1.6 𝑥 10 −19 𝐶 V =5000 𝑉 ∆𝑈=? ∆𝑈=𝑞V=(−1.6 𝑥 10 −19 𝐶)(5000 𝐽 𝐶 ) ∆𝑈=−8.0 𝑥 10 −16 𝐽 Potential energy is lost by electron (and converted into kinetic energy)

Quantifying voltage – part 2 ∆𝑈=−𝑞𝐸𝑑 So, V=−𝐸𝑑 Or, 𝐸=− V 𝑑 Electric field strength is frequently reported in volts / meter

Example An electric field of 525 V/m is desired between two parallel plates that are 11.0 mm apart. How large a voltage should be applied? 𝑑=11.0 𝑚𝑚 =1.1 𝑥 10 −2 𝑚 𝐸 =525 𝑉 𝑚 V=? V=−𝐸𝑑=−(525 𝑉 𝑚 )(1.1 𝑥 10 −2 𝑚) V=−5.8 𝑉 Note the convention: voltage, V volts, V (italicized)

Implications If you know the voltage difference and the distance between two plates, you can calculate the strength of the electric field… And make capacitors And electrocardiograms And oscilloscopes And televisions And mass spectrometers Etc.