Q61P-d A long Iron slab of width w and ... [FREE SOLUTION]

Work and energy with a capacitor: A capacitor with capacitance $C$ has an amount of charge q on one of its plates, in which case the potential difference across the plates is $ΔV = q / C$ (definition of capacitance). The work done to add a small amount of charge dq when charge the capacitor is $dq (ΔV) = dq (q / C)$ . Show by integration that the amount of work required to charge up the capacitor from no charge to final charge Q is $\frac{1}{2} (Q^{2} / C)$ . Since this is the amount of work required to charge the capacitor, it is also the amount of energy stored in the capacitor. Substituting $Q = CΔV$ , we can also express the energy as $\frac{1}{2} C {(ΔV)}^{2}$ .

You connect a $9 V$ battery to a capacitor consisting of two circular plates of radius $0.08 m$ separated by an air gap of $2mm$ , what is the charge on the positive plate?

A long Iron slab of width w and height h emerges from a furnace, as shown in Figure 19.79. Because the end of the slab near the furnace is hot and the other end Is cold, the electron mobility increases significantly with the distance x. The electron mobility is $u = u_{0} + k x$ where u₀is the mobility of the iron at the hot end of the slab. There are n iron atoms per cubic meter, and each atom contributes one electron to the sea of the mobile electron (we can neglect the small thermal expansion of the iron). A steady state conventional current runs through the slab from the hot end towards cold end, and an ammeter (not shown) measures the current to have a magnitude I in amperes. A voltmeter is connected to two locations a distance d apart, as shown. (a) Show the electric field inside the slab at two locations marked with $\times$ . Pay attention to the relative magnitudes of the two vectors that you draw. (b) Explain why the magnitude of the electric field is different at these two locations. (c) At a distance x from the left voltmeter connection, what is the magnitude of the electric field in terms x and the given quantities w, h, d, u₀, k, l, and n ( and fundamental constants)? (d) What is the sign of potential difference displayed on the voltmeter? Explain briefly. (e) In terms of the given quantitiesw, h, d, u₀, k, l, and n and ( and fundamental constants), what is the magnitude of the voltmeter reading? Check your work. (f) What is the resistance of this length of the iron slab?

Consider two capacitors whose only difference is that the plates of capacitor number 2 are closer together than those of capacitor number 1 (Figure 19.56). Neither, capacitors has an insulating layer between the plates. They are placed in two different circuits having similar batteries and bulbs in series with the capacitor.

Show that in the first fraction of a second, the current stays nearly constant (decreases less rapidly) in the circuit with capacitor number 2. Explain your reasoning in detail.

Hint: Show charges on metal plates, and consider the electric fields they produce in the nearby wires. Remember that the fringe field near a plate outside a circular capacitor is approximately-

$(\frac{\frac{Q}{A}}{ε_{o}}) (\frac{s}{2 R})$

More extensive analysis shows that this trend holds true for the entire charging process: the capacitor with the narrower gap ends up with more charge on the plates.

A circuit consists of a battery, whose emf is K, and five Nichrome wires, three thick and two thin as shown in Figure 19.78. The thicknesses of the wires have been exaggerated in order to give you room to draw inside the wires. The internal resistance of the battery is negligible compared to the resistance of the wires. The voltmeter is not attached until part (e) of the problem. (a) Draw and label appropriately the electric field at the locations marked × inside the wires, paying attention to appropriate relative magnitudes of the vectors that you draw. (b) Show the approximate distribution of charges for this circuit. Make the important aspects of the charge distribution very clear in your drawing, supplementing your diagram if necessary with very brief written descriptions on the diagram. Make sure that parts (a) and (b) of this problem are consistent with each other. (c) Assume that you know the mobile-electron density n and the electron mobility u at room temperature for Nichrome. The lengths $(L_{1}, L_{2}, L_{3})$ and diameters $(d_{1}, d_{2})$ of the wires are given on the diagram. Calculate accurately the number of electrons that leave the negative end of the battery every second. Assume that no part of the circuit gets very hot. Express your result in terms of the given quantities $(K, L_{1}, L_{2}, L_{3}, d_{1}, d_{2}, n and u)$ . Explain your work and identify the principles you are using. (d) In the case that $d_{2} ≪ d_{1}$ , what is the approximate number of electrons that leave the negative end of every second? (e) A voltmeter is attached to the circuit with its + lead connected to location B (halfway along the leftmost thick wire) and its - lead connected to location C (halfway along the leftmost thin wire). In the case that $d_{2} ≪ d_{1}$ , what is the approximate voltage shown on the voltmeter, including sign? Express your result in terms of the given quantities $(K, L_{1}, L_{2}, L_{3}, d_{1}, d_{2}, n and u)$ .

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(d) Determining the sign of potential difference displayed on the voltmeter

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