Electrostatics: Coulomb’s Law & Electric Fields

Electric Charges  There are two kinds of charges: positive (+) and negative (-), with the following relationships: Like charges (same sign) repel each other Unlike charges (opposite sign) attract each other

Characteristics of Electric Charge  Electric charge is never created or destroyed – it is conserved  Charge always comes in a multiple of a basic unit: e -, where e = x Coulombs (C) The charge on an electron is x Coulombs A proton has the opposite charge

Electric Charges  Charge always comes in a multiple of that basic unit: q = Ne, where q is charge and N is the number of electrons or protons

Problem One:  A certain static discharge delivers -0.5 Coulombs of electrical charge. How many electrons are in this discharge?

Sample Problem A certain static discharge delivers -0.5 Coulombs of electrical charge. How many electrons are in this discharge?

Problem Two  How much positive charge resides in two moles of hydrogen gas? (H 2 )  How much negative charge?  How much net charge?

Sample Problem 1.How much positive charge resides in two moles of hydrogen gas (H 2 )? 2.How much negative charge? 3.How much net charge?

Transfer of Electric Charge  Charge can be transferred between objects Transfer of charge is almost always due to the transfer of electrons Remember: Atomic nuclei are fixed, but the outer electrons are more easily separated, leaving negative electrons and positive ions

Coulomb’s Law

 When opposite charges (let’s say q 1 and q 2 ) are separated, they are attracted by an electric force (like charges are repelled)  The attractive force can be determined using Coulomb’s Law :  Where q 1 and q 2 are charges, k is the electrostatic constant, and r is the distance between the charges

Electrostatic Constant  A quick note: k = 1/(4πε 0 ) ε 0 is the permittivity of free space ε 0 = 8.85 x C 2 /Nm 2  BUT you don’t need to know that, because you can use k = 9.0 x 10 9 Nm 2 /C 2  By the way, Coulomb’s Law only applies directly to spherically symmetric charges

Coulomb’s Law Example  The radius of a hydrogen atom is 5.29 x m  What is the electric force between a proton and an electron in a hydrogen atom?

 A hydrogen atom has one proton in its nucleus and one electron orbiting the nucleus. The magnitude of the charge of the electron is the same as the magnitude of the charge of the proton and equals 1.60 × 10 −19 C. The magnitude of the electric force is determined by the Coulomb’s Law

Yet Another Problem  A point charge of positive 12.0  C experiences an attractive force of 51 mN when it is placed 15 cm from another point charge. What is the other charge?

Sample Problem A point charge of positive 12.0 μC experiences an attractive force of 51 mN when it is placed 15 cm from another point charge. What is the other charge?

Superposition  Electric force (like ALL forces) is a vector quantity. (don’t you just love geometry?)  If a charge is subjected to forces from more than one other charge, we use VECTOR ADDITION! Yay!  Sometimes that’s called superposition (just so you know)

Practice with Superpositon  What is the force on the 4  C charge?

Sample Problem What is the force on the 4  C charge? y (m)  C-3  C4  C x (m)

The Electric Field  The presence of electric charge modifies empty space. The electric force can act on charged particles without actually touching them (like gravity acts on distant masses)  We say that an “electric field” is created in the space around a charged particle or a configuration of charges

The Electric Field  If a charged particle is placed in an electric field created by other charges, it will experience a force from the field  Sometimes we know about the electric field without knowing about the charge configuration that created it. We can easily calculate the electric force from the field instead of the charges.

Why use fields?  Forces exist only when two or more particles are present  Fields can be calculated for just one particle  Fields exist even if there is no net force  The arrows in a field are NOT VECTORS – they are LINES OF FORCE  Field lines indicate the direction of force on a positive charge placed in the field (opposite for negatives)

Field between charged plates

Calculating Electric Field  The force on a charged particle placed in an electric field can be calculated by:  F = Eq F: Force (N) E: Electric Field (N/C) Q: charge (C)

Field Practice  The electric field in a given region is 4000 N/C pointed North. What is the force exerted on a 400  g styrofoam bead bearing 600 excess electrons when placed in the field?

Sample Problem The electric field in a given region is 4000 N/C pointed toward the north. What is the force exerted on a 400 μg styrofoam bead bearing 600 excess electrons when placed in the field?

More Practice  A proton traveling at 440 m/s in the +x direction enters an electric field of magnitude 5400 N/C directed in the +y direction. Find the acceleration.

Sample Problem A proton traveling at 440 m/s in the +x direction enters an an electric field of magnitude 5400 N/C directed in the +y direction. Find the acceleration.

Spherical Electric Fields  The electric field surrounding a point charge or spherical charge can be calculated by:  E = kq/r 2 E: Electric field (N/C) K: 9x10 9 Nm 2 /C 2 q: Charge (C) r: distance from center of charge q (m)

Superposition with Fields  When more than one charge contributes to the electric field, the resultant field is the vector sum of the electric fields from the individual charges  Remember: Electric field lines are NOT VECTORS, but can be used to find the direction of the electric field vectors.

Yay More Practice  A particle bearing -5.0  C is placed at -2.0 cm, and a particle bearing 5.0  C is placed at 2.0 cm. What is the field at the origin?

Sample Problem A particle bearing -5.0 μC is placed at -2.0 cm, and a particle bearing 5.0 μC is placed at 2.0 cm. What is the field at the origin?

Electrostatics: Electric Potential & Potential Energy; Energy Conservation & Potential Equipotential Lines

Electric Potential Energy  Electric potential energy ( U E ) – energy contained in a configuration of charges Increases when configuration becomes less stable Decreases when configuration becomes more stable Unit: Joule

Electric Potential Energy  Work must be done on the charge to increase electric potential energy  For a positive test charge to be moved upward a distance d, the electric force does negative work  The electric potential energy has increased and ΔU is positive

Work and Energy  If a negative charge is moved upward a distance d, the electric force does positive work.  The change in the electric potential energy (ΔU) is negative

Electric POTENTIAL  Electric potential (commonly called VOLTAGE) is related to both electric potential energy, and the electric field Units are the Volt, where 1V = 1 J/C  Change in potential energy is directly related to change in voltage: ΔU = qΔV ○ Δ U is the change in electrical PE (unit: J) ○ q is the charged moved (unit: C) ○ Δ V is the potential difference (V)

Electric Potential & Potential Energy  All charges will spontaneously go to lower potential energies if allowed to move – they try to decrease U E  Positive charges like to DECREASE their potential (Δ V < 0)  Negative charges like to INCREASE their potential (Δ V > 0)

Practice #1  A 3.0  C charge is moved through a potential difference of 640 V. What is its change in potential energy?

Sample Problem A 3.0 μC charge is moved through a potential difference of 640 V. What is its potential energy change?

Electric Potential in Uniform Fields  The electric potential is related to a uniform electric field:  Δ V = -Ed Δ V is the change in electric potential (V) E is a constant electric field strength d is the distance moved (m)

Practice Problem #2  An electric field is parallel to the x- axis. What is the magnitude and direction of the electric field if the potential difference between x = 1.0m and x = 2.5m is found to be +900V?

Sample Problem An electric field is parallel to the x-axis. What is its magnitude and direction if the potential difference between x =1.0 m and x = 2.5 m is found to be +900 V?

Charges on Conductors  Excess charges reside on the surface of a charged conductor  If excess charges were found inside a conductor, they would repel one another until the charges were as far from each other as possible – on the surface

Electric Fields & Conductors  Electric field lines are more dense near a sharp point – this means the field is more intense in these regions  Lightning rods have a sharply pointed tip  During an electrical storm, the electric field at the tip becomes so intense that charge is given off into the atmosphere, discharging the area near a building at a steady rate and preventing sudden blasts of lightning

Electric Fields and Conductors  The electric field inside a conductor MUST be zero  If a conductor is placed in an electric field, the charges polarize to nullify the external field

Conservation of Energy  Conservative forces conserve energy – mechanical energy changes from one form to another  When only the conservative electrostatic force is involved, a charged particle released from rest in an electric field will transform potential energy into kinetic energy

Practice #5  A proton is accelerated through a potential difference of -2,000 V. What is its change in potential energy?  How fast will it be moving if it started at rest?

Sample Problem If a proton is accelerated through a potential difference of 2,000 V, what is its change in potential energy? How fast will this proton be moving if it started at rest?

#6  A proton at rest is released in a uniform electric field. How fast is it moving after it travels through a potential difference of V?

Sample Problem A proton at rest is released in a uniform electric field. How fast is it moving after it travels through a potential difference of V? How far has it moved?

Electric Potential Energy  For spherical/point charges:  U = kq 1 q 2 /r U is electrical PE (J) K is 9 x 10 9 Nm 2 /C 2 q 1 and q 2 are charges (C) r is the distance between centers (m)

#7  How far must the point charges q 1 =  C and q 2 =  C be separated for the electric potential energy of the system to be -126 J?

Absolute Electric Potential  For a spherical/point charge, the electric potential can be calculated by:  V = kq/r V is potential (V) k = 9 x 10^9 q is charge (C) r is distance from the charge (m)

#8  The electric potential 1.5 m from a point charge q is +2.8 x 10 4 V. What is the value of q?

More about Electric Fields & Electric Potential  E = -ΔV/d  The electric field points in the direction of decreasing electric potential  The electric field is always perpendicular to the equipotential surface

E and Equipotential are Perpendicular!  No work is done when a charge is moved perpendicular to an electric field  If no work is done, there is no change in potential  Potential is constant in a direction perpendicular to the electric field

Equipotential Surfaces

No. NINE  Draw field lines for the charge configuration below. The field is 600 V/m, and the plates are 2m apart. Label each plate with its proper potential, and draw and label 3 equipotential surfaces between the plates. (Ignore edge effects)

Sample Problem Draw field lines for the charge configuration below. The field is 600 V/m, and the plates are 2 m apart. Label each plate with its proper potential, and draw and label 3 equipotential surfaces between the plates. You may ignore edge effects

Ten  Draw a negative point charge of –Q and its associated electric field. Draw 4 equipotential surfaces such that ΔV is the same between the surfaces, and draw them at the correct relative locations. What do you observe about the spacing between the surfaces?

Sample Problem Draw a negative point charge of -Q and its associated electric field. Draw 4 equipotential surfaces such that  V is the same between the surfaces, and draw them at the correct relative locations. What do you observe about the spacing between the equipotential surfaces?