A conducting loop is fixed in a region with a magnetic field directed perpendicular to the plane of the loop. The induced electromotive force (EMF) \(\mathcal{E}\) in the loop is measured as a function of time \(t\) and is shown in the graph. At time \(t = 0\), the magnetic flux through the loop is zero (\(\Phi_B = 0\)). Which of the following graphs best represents the magnetic flux \(\Phi_B\) through the loop as a function of time \(t\)?

A rigid circular wire loop of radius \(a\) and total resistance \(R\) lies fixed in a plane perpendicular to a spatially uniform magnetic field. The magnitude of the magnetic field changes with time \(t\) according to \(B(t) = B_0 \left(1 + \dfrac{t}{\tau}\right)^3\), where \(B_0\) and \(\tau\) are positive constants. Which of the following expressions represents the total electric charge \(Q\) that flows past a point in the loop between \(t = 0\) and \(t = \tau\)?

A flat rectangular loop of wire with length \(L\), width \(w\), and resistance \(R\) moves in the \(+x\)-direction at constant speed \(v_0\). At time \(t = 0\), the front edge of the loop enters a magnetic field region extending from \(x = 0\) to \(x = 2L\). Within this region, the magnetic field is directed into the page with spatially varying magnitude \(B(x) = B_0 \dfrac{x}{L}\), where \(B_0\) is a positive constant, and \(B = 0\) elsewhere. Defining counterclockwise current as positive, the induced current \(I(t)\) as a function of time increases linearly from \(0\) to \(+I_0\) for \(0 \le t \le T\), remains constant at \(+I_0\) for \(T \le t \le 2T\), and jumps to \(-I_0\) at \(t = 2T\) before rising linearly to \(0\) at \(t = 3T\), where \(T = \dfrac{L}{v_0}\) and \(I_0 = \dfrac{B_0 w v_0}{R}\). Which of the following claims correctly explains the physical origin of a feature in the \(I(t)\) graph?

An ideal circuit consists of an inductor of inductance \(L\) connected in series with a capacitor of capacitance \(C\). At time \(t = 0\), the capacitor carries an initial charge \(Q_0\) and the current in the circuit is zero. Which of the following graphs best represents the charge \(q(t)\) on the capacitor as a function of time \(t\)?

A square loop of wire with side length \(L\) lies in a region containing a non-uniform magnetic field \(\vec{B} = B_0 \left(\dfrac{x}{L}\right) \hat{k}\), where \(B_0\) is a positive constant and \(x \ge 0\). One edge of the loop is fixed along the \(y\)-axis from \(y = 0\) to \(y = L\). The loop is tilted about the \(y\)-axis by an angle \(\theta\) relative to the \(xy\)-plane. What is the ratio of the magnetic flux through the loop when \(\theta = 60^\circ\) to the magnetic flux through the loop when \(\theta = 0^\circ\)?

A horizontal circular ring of radius \(r\), mass \(m\), and electrical resistance \(R\) falls vertically under the influence of gravity through a region with a non-uniform vertical magnetic field. The vertical component of the magnetic field varies linearly with height \(z\) according to \(B_z(z) = B_0 + bz\), where \(B_0\) and \(b\) are positive constants, and \(z\) is measured upward. Air resistance is negligible. Which of the following expressions correctly represents the magnitude of the ring’s terminal velocity \(v_T\)?

A long, straight cylindrical wire of radius \(R\) carries a steady total current \(I\) distributed uniformly across its circular cross section. The permeability of free space is \(\mu_0\). Which of the following expressions represents the total magnetic energy stored per unit length inside the volume of the wire?

A circular region of radius \(R\) in the \(xy\)-plane contains a time-dependent, non-uniform magnetic field directed perpendicular to the plane. Inside the region (\(r \le R\)), the magnitude of the magnetic field is given by \(B(r,t) = C t \left(\dfrac{r}{R}\right)^2\), where \(C\) is a positive constant and \(r\) is the radial distance from the central axis. Outside the region (\(r > R\)), the magnetic field is zero. Which of the following expressions gives the magnitude of the induced electric field \(E(r)\) as a function of distance \(r\) from the central axis inside the region (\(r < R\))?

In the circuit shown, an ideal battery with potential difference \(V_0\) is connected to three resistors (\(R_1 = R\), \(R_2 = 2R\), \(R_3 = R\)), an ideal inductor \(L\), and a switch \(S\). The switch \(S\) has been closed for a long time. At time \(t = 0\), switch \(S\) is opened. Which of the following expressions represents the magnitude of the current \(I(t)\) through resistor \(R_2\) as a function of time \(t\) for \(t \ge 0\)?

A circuit consists of an ideal battery of constant emf \(\varepsilon\), a resistor of resistance \(R\), an inductor of inductance \(L\), and an open switch connected in series. At time \(t = 0\), the switch is closed. At the instant when the current in the circuit reaches half of its maximum steady-state value, what fraction of the total energy delivered by the battery up to that time is stored in the magnetic field of the inductor?