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A capacitor being charged has a current carrying charge to and away from the plates. In the next chapter we will define current to be $dQ/dt$ the rate of charge flow. What is the current to a $10\mathrm{μ}\mathrm{F}$ capacitor whose voltage is increasing at the rate of $2.0\mathrm{V}/\mathrm{s}$?

Figure Q26.7 shows an electric field diagram. Dashed lines 1 and 2 are two surfaces in space, not physical objects.

a. Is the electric potential at point a higher than, lower than, or equal to the electric potential at point b? Explain.

b. Rank in order, from largest to smallest, the magnitudes of the potential differences $∆V_{ab},∆V_{cd},\mathrm{and}∆V_{ef}$.

c. Is surface 1 an equipotential surface? What about surface 2? Explain why or why not.

What are the magnitude and direction of the electric field at the dot in Figure EX26.7?

The current that charges a capacitor transfers energy that is stored in the capacitor's electric field. Consider a $2.0\mathrm{μ}\mathrm{F}$capacitor, initially uncharged, that is storing energy at a constant $200\mathrm{W}$rate. What is the capacitor voltage role="math" localid="1648645414866" $2.0\mathrm{μ}s$ after charging begins?

A typical cell has a membrane potential of $-70\mathrm{mV}$, meaning that the potential inside the cell is $70\mathrm{mV}$ less than the potential outside due to a layer of negative charge on the inner surface of the cell wall and a layer of positive charge on the outer surface. This effectively makes the cell wall a charged capacitor. Because a cell's diameter is much larger than the wall thickness, it is reasonable to ignore the curvature of the cell and think of it as a parallel-plate capacitor. How much energy is stored in the electric field of a $50\mathrm{μ}m$diameter cell with a $7.0nm$thick cell wall whose dielectric constant is $9.0$?

A nerve cell in its resting state has a membrane potential of $-70\mathrm{mV}$, meaning that the potential inside the cell is $70\mathrm{mV}$ less than the potential outside due to a layer of negative charge on the inner surface of the cell wall and a layer of positive charge on the outer surface. This effectively makes the cell wall a charged capacitor. When the nerve cell fires, sodium ions,${\mathrm{Na}}^{+}$, flood through the cell wall to briefly switch the membrane potential to $+40\mathrm{mV}$. Model the central body of a nerve cell-the soma-as a $50\mathrm{μ}m$diameter sphere with a $7.0-\mathrm{nm}$-thick cell wall whose dielectric constant is 9.0. Because a cell's diameter is much larger than the wall thickness, it is reasonable to ignore the curvature of the cell and think of it as a parallel-plate capacitor. How many sodium ions enter the cell as it fires?

Derive Equation 26.33 for the induced surface charge density on the dielectric in a capacitor.

A vacuum-insulated parallel-plate capacitor with plate separation $d$has capacitance $C_0$. What is the capacitance if an insulator with dielectric constant $\mathrm{κ}$and thickness $d/2$is slipped between the electrodes without changing the plate separation?

In Problems 75 through 77, you are given the equation(s) used to solve a problem. For each of these, you are to: a. Write a realistic problem for which this is the correct equation (s). b. Finish the solution to the problem.

$2az\mathrm{V}/\mathrm{m}=-\frac{dV}{dz}$whereas a is a constant with units of$V/m^{2}V(z=0)=10V$.

In Problems 75 through 77, you are given the equation(s) used to solve a problem. For each of these, you are to

a. Write a realistic problem for which this is the correct equation (s).

b. Finish the solution to the problem.

76). $400\mathrm{nC}=(100\mathrm{V})C$