At a given instant, a 2.4-A current flows in the wires connected to a parallel-plate capacitor. What is the rate at which the electric field is changing between the plates if the square plates are 1.60 cm on a side?

(III) A long straight wire and a small rectangular wire loop lie in the same plane, Fig. 30–25. Determine the mutual inductance in terms of 𝓁₁, 𝓁₂, and w. Assume the wire is very long compared to 𝓁₁, 𝓁₂, and w, and that the rest of its circuit is very far away compared to 𝓁₁, 𝓁₂, and w.

(II) If the solenoid in Fig. 29–47 is being pulled away from the loop shown, in what direction is the induced current in the loop? Explain.

A coil has 3.25-Ω resistance and 440-mH inductance. If the current is 3.00 A and is increasing at a rate of 3.15 A/s, what is the potential difference across the coil at this moment?

(II) Part of a single rectangular loop of wire with dimensions shown in Fig. 29–49 is situated inside a region of uniform magnetic field of 0.650 T. The total resistance of the loop is 0.250 Ω. Calculate the force required to pull the loop from the field (to the right) at a constant velocity of 3.40 m/s. Neglect gravity.

(III) A toroid has a rectangular cross section as shown in Fig. 30–26. Show that the self-inductance is

$L=\frac{{\mu }_0{N}^{2}h}{2\pi }\ln \frac{r_2}{r_1}$

where N is the total number of turns and r₁, r₂ and h are the dimensions shown in Fig. 30–26. [Hint: Use Ampère’s law to get B as a function of r inside the toroid, and integrate.]

The magnetic field inside an air-filled solenoid 38.0 cm long and 2.10 cm in diameter is 0.720 T. Approximately how much energy is stored in this field?

What is the energy density at the center of a circular loop of wire carrying a 19.0-A current if the radius of the loop is 28.0 cm?

(II) For the toroid of Fig. 30–26, determine the energy density in the magnetic field as a function of r(r₁ < r < r₂) and integrate this over the volume to obtain the total energy stored in the toroid, which carries a current I in each of its N loops.

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(III) Determine the emf induced in the square loop in Fig. 29–52 if the loop stays at rest and the current in the straight wire is given by I(t) = (15.0 A) sin (2200 t) where t is in seconds. The distance a is 12.0 cm, and b is 15.0 cm.

(II) (a) Determine the energy stored in the inductor L as a function of time for the LR circuit of Fig. 30–6a. (b) After how many time constants does the stored energy reach 99.9% of its maximum value?

(II) (a) In Fig. 30–28, assume that the switch has been in position A for sufficient time so that a steady current I₀ = V₀/R flows through the resistor R. At time t = 0, the switch is quickly switched to position B and the current decays through resistor R' (which is much greater than R) according to . Show that the maximum emf ε_max induced in the inductor during this time period is (R'/R)V_o. (b) If R' = 45R and V_o = 145 V, determine ε_max. [When a mechanical switch is opened, a high-resistance air gap is created, which is modeled as R' here. This Problem illustrates why high-voltage sparking can occur if a current-carrying inductor is suddenly cut off from its power source. The very high voltage can produce an electric field great enough to ionize atoms of air, which emit light when electrons recombine with the ions.]