PHYS1200Spring2024_Class04_Potential(2)

PHYS-1200/1250 Lab Manual Name_________________________ Section__________ 23A – Electric Potential and Electric Potential Energy Equipment: Pencil, paper, and brain. Key Theoretical Ideas: Reading: Young and Freedman Chapter 23.1-23.5 1) Write the relationship for the force on a point charge q test due to an electric field E ( x ): ⃗ F ___________ 2) Find the dot product ⃗ F ⋅ d ⃗ s for ⃗ E = ¿ (3 ^ i + 2 ^ j ) N C and q test = +1nC and d ⃗ s = ¿ dx ^ i . ⃗ F ⋅ d ⃗ s `=_______________(units)_______ 3) Find the change in potential energy for the particle above if it is moved from x = 2 to x = 7 m. __________________(units)_______ (Aside: If the path and/or the field are curved, then we need to perform a more careful integration, which we won’t address in this course.) 4) Assume that the electric field in a region of space is ⃗ E = ( 3 1 m ) x ^ i N C and find the difference in potential energy if a charge q test = +1 n C is moved from x = 2 m to x = 7 m with d ⃗ s = ¿ dx ^ i . Write your answer on the line but include sufficient steps in your solution so the grader can follow what you did. ΔU = ¿ ¿ __units For the situation above, we separately specified the field, the test charge, and the position. It will be very useful to us later if we create a new quantity, called “electric potential ,” that separates the effects of the choice of the test charge from the field and position. 1 Relationships Between Electric Field, Potential Energy, and Electric Potential We know from PHYS-1100 that moving an object with a conservative force gives rise to a change in potential energy, . U f − U i =− ∫ initial final ⃗ F ⋅ d ⃗ s Eq. 23a (The change in potential energy is equal to the negative of the work done on the object.) You studied this for forces due to springs and gravity. This idea can be applied to electric forces as well. Note that Eq. 23a involves a dot product between the force and the steps (d s ) along the path of the

PHYS-1200/1250 Lab Manual Name_________________________ Section__________ ΔV ≡ ΔU q test = − 1 q test ∫ initial final q test ⃗ E ⋅ d ⃗ s =− ∫ initial final ⃗ E ⋅ d ⃗ s Eq 23b 5) Assume that the electric potential in a region of space is given by V = ( [ 3 m ] x + [ 2 m 2 ] y 2 + 1000 ) V . a) Find the x-component of the electric field at the point (4,4,0): E x (4, 4, 0) =______________. b) Find the y-component of the electric field at the point (4,4,0): E y (4, 4, 0) =______________. c) Find the z-component of the electric field at the point (4,4,0): E z (4, 4, 0) =______________. Calculating the Electric Potential from Known Charges 6) For a given situation, calculate a few observational points for electric potential to determine the geometry of the configuration. a) Calculate the potential at the point P = (0,3,0) due to a +1 nC point charge at P’= (0,1,0). V =____________V(volts) b) Calculate the potential at the point P = (2, 1, 0) due to a +1 nC point charge at P’= (0,1,0). V =____________V(volts) c) How does the potential depend on the direction of the measuring point from a point source? 2 The difference in electric potential between two points is what we measure when we use a voltmeter. Unfortunately the symbol for the MKS unit of voltage is V (volts) and the symbol for electric potential is also V. Check for context. If we know the electric potential ( V ( x, y, z ))as a function of position, then we can find the electric field components E x , E y , E z by taking derivatives: E x ( x , y , z ) = − ∂V ( x , y ,z ) ∂ x ; E y ( x , y , z ) = − ∂V ( x , y, z ) ∂ y ; E z ( x , y ,z ) = − ∂V ( x, y ,z ) ∂ z The electric potential at point P = ( x, y, z ) due to a charge q at point P ’ = ( x’, y’, z’ ) is: Eq. 23d The potential at a point due to a collection of charges is the sum of the potentials from each of the charges. (equations 23.15 and 23.16 in Young and Freedman). Note: For finite charge distributions we usually take the potential at infinite distance from the charge to be zero. Potentials are usually measured with respect to this potential.

PHYS-1200/1250 Lab Manual Name_________________________ Section__________  Click on the pencil again to draw an equipotential that is slightly to the left of midway.  There are two separated equipotential surfaces with the value 25V. Find and click on the pencil to draw them on the screen.  Find the equipotential(s) for 15V and click the pencil to draw them on the screen. 6) Make a sketch below of the charges and equipotentials (including the surfaces above) for the two positive charges. Note that the electric field on an equipotential surface is always perpendicular to that surface.  Drag the second positive charge back to the box at the bottom of the screen. Drag a -1 nC charge to a point 2 meters to the right of the first charge. 7) Sketch the charges and equipotentials for several equipotentials including +5 V, +2 V, +1 V, and 0 V. 8) A top view of two equipotentials in a certain region of space are sketched to the right. Describe the region(s) in which the electric field has the greatest magnitude. (e.g.- “At the center of the smaller circle.” “To the left of center and between the two circles.”) Explain your answer. 9) The electric field vector is always perpendicular to any equipotential surface. Explain why. 10) Under electrostatic equilibrium (no charge flow) the interior of a conductor has zero electric field. Explain how this leads to the conclusion that the potential within a conductor is a constant. 4

PHYS-1200/1250 Lab Manual Name_________________________ Section__________ 23C – Experiment: Electric Potential and Electric Field  You will use the multimeter to measure the difference in electric potential between various points on the conducting paper.  Set up the measurement system as indicated below. Solid lines are wires.  Set the power supply voltage to 10 V. Connect the two black wires to the same silvered point on the paper. The brass paperclips should be used with the alligator clips to prevent them from ripping the paper. (Banana wires can be piggybacked into one another.) Connect the red wire from the power supply to the opposite dot on the paper. The red wire from the DMM will be used as a variable position probe .  Set the multimeter to measure DC Voltage. (Indicated by a V .)  Drag the probe tip gently around the paper to find the shape of the equipotential for 2V, 4V, 6V, and 8V. Your voltmeter measures the difference in potential between one probe tip and the other. Discuss what this means with your classmates and facilitators. 5 Equipment: 1 rectangular conducting black paper, 1 DC power supply w/power cable, 2 red banana cables, 2 black banana cables, 2 brass paperclips, 2 alligator clips, 1 digital multimeter, 1 ruler for measuring position.

PHYS-1200/1250 Lab Manual Name_________________________ Section__________ 2. Measure the potential difference between the left hand source dot and the probe tip as a function of probe position along a straight line from the center of one contact to the other and enter it in the second column. Measure in a direction such that the electrical potential (V) decreases from x = 3 cm to x = 20cm. 7 Position (cm) Measured Potential (V) Midpoint (cm) Δ x (m) Δ V (V) Field -Δ V /Δ x (V/m) 3 4.5 6 7.5 9 10.5 12 13.5 15 16.5 18 19.5 21 22.5 24 25.5 27

PHYS-1200/1250 Lab Manual Name_________________________ Section__________ 3. Plot the potential as a function of position on the grid below. If you wish, you can do this in the word version of this file by dragging the points to appropriate positions on the grid. Don’t forget to add a scale. Test if the behavior is linear by dragging the line to make an approximate “best fit”. 4. Calculate the electric field between adjacent measured points. Add it to the last column of the table. 5. Plot the average field as a function of position on the grid below. a) Is the field constant across the paper? _____________ b) Explain why you think the field is constant (or not) using the concepts of the potential near point charges. 8

PHYS-1200/1250 Lab Manual Name_________________________ Section__________ Equipotentials and Conductors 9) For the cases sketched below, indicate the directions of the electric field near each of the surfaces. Shaded regions are conducting. In each case, the left conductor is at positive potential with respect to the right one. Case a) Case b) 10 Important Conclusions about Electric Potential : When all charges in a system are at rest, the entire volume of a conductor must be an equipotential . This is because if there is a non-zero electric field in the conductor, then charge in the conductor will be accelerated and flow. There is no potential difference between two points between which the field is zero. When all charges are at rest, the electric field outside of a conductor must be perpendicular to the surface of the conductor . This is because the surface of the conductor is an equipotential by the argument above.

PHYS-1200/1250 Lab Manual Name_________________________ Section__________ EXTRA CREDIT (3 pts) 10) Consider a situation in which there are two conducting spheres that are far apart but connected by a long thin conducting wire. One sphere of radius 1.0 mm has charge +3.0 nC. The other sphere has radius R = 3.0 mm, but the charge is not given. You are asked to find the charge on the R= 3.0 mm sphere theoretically. The questions below may help you to find the solution. a) Give an argument for why the surfaces of the small and large spheres are at the same potential. b) Given the charge on the small sphere find the potential at the surface of the small sphere. c) Given the potential, find the charge on the large sphere. 11