Magnetic Flux：練習問題

A long solenoid of radius \(R\) has \(n\) turns per unit length and carries a time-varying current \(I(t) = I_0 \sin(\omega t)\). A small circular coil of radius \(r < R\) and resistance \(R_{coil}\) is placed inside the solenoid with its axis parallel to the solenoid's axis. What is the maximum induced current in the small coil?

A coaxial cable consists of an inner solid conductor of radius \(a\) and an outer thin cylindrical shell of radius \(b\). They carry equal and opposite currents \(I\) distributed uniformly. What is the magnetic energy stored per unit length in the region between the conductors (\(a < r < b\))?

A thin conducting rod of length \(L\) rotates about an axis perpendicular to its length through one of its ends with a constant angular velocity \(\omega\). The rotation occurs in a uniform magnetic field \(B\) directed parallel to the axis of rotation. Find the potential difference between the two ends of the rod.

A square loop of side \(L\) and resistance \(R_{res}\) is pulled at a constant velocity \(v\) out of a region of uniform magnetic field \(B\) which is perpendicular to the plane of the loop. The force required to maintain this constant velocity is:

A rectangular loop of wire with dimensions \(a\) and \(b\) lies in the \(xy\)-plane. A non-uniform magnetic field is given by \(\mathbf{B} = B_0 \left( 1 + \frac{y}{b} \right) \mathbf{\hat{k}}\). If the loop moves with a constant velocity \(\mathbf{v} = v_0 \mathbf{\hat{j}}\), determine the magnitude of the induced electromotive force (emf) in the loop at the instant its trailing edge is at \(y = 0\).

A rectangular wire loop of resistance \(R\), mass \(m\), and width \(w\) is dropped from rest into a region with a uniform horizontal magnetic field \(B\) that is perpendicular to the plane of the loop. Determine the terminal velocity \(v_t\) of the loop as it enters the magnetic field, assuming the top edge is still outside the field.

A rectangular loop of wire with dimensions \(w\) and \(l\), and total resistance \(R_{res}\), is being pulled at a constant velocity \(v\) out of a region of uniform magnetic field \(B\) directed into the page. The loop is oriented such that its width \(w\) is perpendicular to the direction of motion.
(a) Calculate the magnitude of the induced EMF in the loop as it leaves the field.
(b) Derive an expression for the external force \(F_{ext}\) required to maintain the constant velocity.
(c) Show that the rate of work done by the external force equals the rate of thermal energy dissipation in the loop.

A conducting rod of mass \(m\) and length \(L\) slides without friction down two vertical conducting rails connected at the top by a resistor \(R\). A uniform magnetic field \(B\) is directed horizontally, perpendicular to the plane of the rails. The rod starts from rest.
(a) Determine the terminal velocity \(v_t\) reached by the rod.
(b) Find the velocity of the rod as a function of time \(v(t)\).
(c) Calculate the total power dissipated in the resistor when the rod reaches terminal velocity.