Preview File Edit View EDV 60 Tools Window Help Discussion 4 worksheet.pdf Page i of a A neuron (pictured) is a body cell that creates and transmits electrical signals. To do this, it uses chemical processes to maintain an imbalance of electric charges on either side of its cell membrane, creating an electric field. In this problem, we will estimate the amount of charge on the membrane. Neurons come in a range of sizes, but a typical one may have a cell body of 20 um = 2 X 10'5 m across. The thickness of the membrane is the same for most neurons: about 5 nm = 5 X 10'9 m, The electric field within a resting nerve membrane is about 1.4 X 107 N/C (assuming the space within the membrane is empty), while the electric field outside the membrane is approximately 0. We could model a 1 um X 1 pm section of the cell membrane as a pair of infinite parallel charged plates with one plate representing the inner edge of the membrane and the other representing the outer edge of the membrane. a. Is it appropriate to model this section of the membrane as infinite plates? Why or why not? We'll assume that one "plate" (side ofthe membrane) is positively charged and the other is negatively charged, as shown in the picture. In all three regions (to the left of the plates, to the right of the plates, and between the plates), draw and label vectors representingthe electric field ofthe positive plate and the electric field of the negative plate. \"Dawn? (POIQEW C Tue Oct 3 11:37 AM 6 Preview File Edit View 60 Tools Window Help Q E (r3) >3 E} 6? Q 8 C TueOct31'l:37AM ED v Discussion 4 worksheet.pdf Page 2 ol A If the net electric field outside the membrane is 0, what does this tell you about the electric field created by each plate? What can you infer about the surface charge density of each plate? Write an expression for the net electric field between the plates, and use it to calculate the surface charge density 17 on each of the plates' Suppose that a sodium ion, Na diffuses across the membrane. When it is between the two sides of the membrane, what is the force that the ion experiences? (9 = 1.602 X 10'1'J C) Modeling the neuron cell body as a sphere with a 20 um diameter, what is the total magnitude of charge on each side of the cell membrane? (We can neglect the distance between the two sides of the membrane when calculating the surface area - why?) no leaosaianewn @639. xam. L woowuv?e:e Preview File Edit View Go Tools Window Help Q Tue Oct 3 11:37 AM Discussion 4 worksheet.pdf Page 3 of 4 Q Search Discussion 4 workshe... g. Keeping the charge density the same, what would be the total amount of charge on each side of the cell membrane if the diameter of the cell were doubled to 40 um? 2. We've studied the electric fields created by point charges, line charges, surface charges, and charged spherical shells. In addition to the charged spherical shell, another three-dimensional distribution of charge is the uniformly charged solid sphere: a sphere where the total charge is uniformly distributed throughout the entire volume (instead of just on the sphere's surface). The volume charge density, p, is the total charge of the sphere divided by the sphere's volume. Like the charged spherical shell, the electric field outside the sphere looks like the field of a point charge (equal to the total charge of the sphere). We can use dimensional analysis to determine the electric field inside the solid sphere, as a function of the distance from the center (the radius r). Of course our equation for the electric field inside the sphere will include an electric constant k. Dimensional analysis doesn't tell us about any numerical factors, but we can guess that there's probably a = It because we're dealing with the volume of a sphere. All that's left is to figure out how the electric field would depend on the volume charge density p and the radius r. a. What are the dimensions of electric field? 3 b. What are the dimensions of the volume charge density? +Preview File Edit View Go Tools Window Help Q Tue Oct 3 11:37 AM Discussion 4 worksheet.pdf Page 3 of 4 . Q Search Discussion 4 workshe... 2. We've studied the electric fields created by point charges, line charges, surface charges, and charged spherical shells. In addition to the charged spherical shell, another three-dimensional distribution of charge is the uniformly charged solid sphere: a sphere where the total charge is uniformly distributed throughout the entire volume (instead of just on the sphere's surface). The volume charge density, p, is the total charge of the sphere divided by the sphere's volume. Like the charged spherical shell, the electric field outside the sphere looks like the field of a point charge (equal to the total charge of the sphere). We can use dimensional analysis to determine the electric field inside the solid sphere, as a function of the distance from the center (the radius r) Of course our equation for the electric field inside the sphere will include an electric constant k. Dimensional analysis doesn't tell us about any numerical factors, but we can guess that there's probably a = It because we're dealing with the volume of a sphere. All that's left is to figure out how the electric field would depend on the volume charge density p and the radius r. a. What are the dimensions of electric field? b. What are the dimensions of the volume charge density? 3 c. Using all of the above information, write down an equation for the electric field inside the sphere. How does it depend on the radius r? +Preview File Edit View Go Tools Window Help Q Tue Oct 3 11:38 AM Discussion 4 worksheet.pdf Page 4 of 4 . Q Search Discussion 4 workshe... 3. Three infinite charged plates are placed on an x-axis as shown, with the middle plate located at the origin. Their surface charge densities are indicated as multiples of o, with o = 2 x 10-6 C/m2. 4 x (um) -80 -60 -20 20 60 80 to -30 +20 On the graph below, plot the electric field along the x-axis as a function of the position x. Show any calculations below. Your graph should indicate both the electric field's magnitude and direction (think about how we indicate direction along a single axis). You will need to choose an appropriate scale for the vertical axis. 3 + x (um) 80 -60 -40 -20 20 40 60 80 4 +