Question

A metal sphere with radius ra is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius rb. There is charge +qon the inner sphere and charge −q on the outer spherical shell. Take V to be zero when r is infinite.Use the equation Er=−∂V∂r and the result from part (b) to find the electric field at any point between the spheres (ra

Answer 1

The electric field at any point between inner sphere and the inner surface of the outer shell with charges +q and -q, respectively, is calculated using Gauss's Law and is equal to q/(4πε0r2), directed radially outward.

Step-by-step explanation:

The student is asking about the electric field within a spherical cavity formed by a metal sphere of radius ra and a concentric hollow metal spherical shell of radius rb. The sphere has a charge +q, and the shell has a charge −q. To find the electric field at any point between the spheres, we first consider the known result that the charge on a conducting sphere distributes itself on the surface. Consequently, the electric field inside a conductor is zero. We can then use Gauss's Law, which states that the electric field E(r) times the surface area of a Gaussian surface equals the charge enclosed divided by the vacuum permittivity ε0.

Inside the cavity (where ra < r < rb), the enclosed charge is +q, thus the electric field E(r) at distance r from the center is identical to the field produced by a point charge +q at the center. By applying Gauss's Law, we find that E(r) = q/(4πε0r2) and is directed radially outward from the center. This field exists only between the inner sphere and the inner surface of the outer shell, as inside the metal of the shell, the electric field is zero.

Answer 2

To prove that a circuit is a causal ZOH circuit, compare its unit impulse response delayed by t/2 seconds to the theoretical response of a ZOH. Circuit analysis using phasor diagrams and time constants will reveal the transient and steady-state behaviors needed for verification.

Step-by-step explanation:

The student is asking to demonstrate that a given circuit is a realization of a causal Zero-Order Hold (ZOH) circuit by comparing its unit impulse response h(t) to a known equation (not explicitly provided in the question) which has been delayed by t/2 seconds to ensure causality.

To show this, one would typically start with the Laplace transform of the system's differential equation, find the impulse response H(s), and then apply the inverse Laplace transform to obtain h(t).

This response would then be examined to see if, when delayed by t/2 seconds, it matches the form of the causal ZOH unit impulse response.

In the context of circuit analysis, understanding the behavior of the circuit components and their relationships using phasor diagrams can be invaluable.

To illustrate, when considering an LR or RC circuit, the phasor representing UR(t) is aligned with i(t), while the phasor for UL(t) or vc(t) is out of phase by π/2 radians.

Analyzing how these phasors interact at various frequencies, especially resonance, and under different conditions, such as after a switch is thrown, sheds light on the transient and steady-state behaviors of the system which are key to understanding ZOH response.

Transient behavior is usually characterized by an exponential approach to a final value, described by the time constant τ of the circuit, which is often graphically represented and easy to analyze.

To show that the circuit is a causal ZOH circuit, one would compare these theoretical insights with the actual response of the circuit to an impulse signal, verifying that the time-domain response aligns with the characteristics of a ZOH system.