## Step 1: Understanding the Physical Situation The solenoid creates a magnetic field \( B \) along its axis, which passes through the loop \( L \). The bead of charge \( q \) moves on the loop (radius \( a \)), with initial velocity \( v_1 \) at \( t= \). The solenoid current changes from \( I_1 \) to \( I_2 \) over time \( \tau \), causing the magnetic field to change from \( B_1 \) to \( B_2). **Explanation:** A changing magnetic field through loop \( L \) induces an electromotive force (EMF) in the loop (Faraday's Law). This EMF will exert a tangential electric force on the charged bead, changing its velocity. --- ## Step 2: Applying Faraday's Law and Newton's 2nd Law **Faraday’s Law:** The induced EMF around the loop is \[ \mathcal{E} = -\frac{d\Phi_B}{dt} \] where \(\Phi_B = B \cdot A = B \cdot \pi a^2\). So, \[ \mathcal{E} = -\pi a^2 \frac{dB}{dt} \] This EMF causes a tangential electric field \( E = \frac{\mathcal{E}}{2\pi a} \), which exerts a force \( F = qE \) on the bead: \[ F = q \cdot \left(-\frac{a}{2} \frac{dB}{dt}\right) \] By Newton’s 2nd Law: \[ F = m \frac{dv}{dt} \] \[ m \frac{dv}{dt} = -\frac{q a}{2}\frac{dB}{dt} \] Integrating both sides over time from to \(\tau\): \[ \int_{v_1}^{v_2} m\,dv = -\frac{q a}{2} \int_{B_1}^{B_2} dB \] \[ m(v_2-v_1) = -\frac{q a}{2}(B_2 - B_1) \] \[ v_2 = v_1 - \frac{q a}{2m} (B_2 - B_1) \] **Explanation:** The bead's velocity changes due to the induced electric field from the changing magnetic field. The direction/sign of the change depends on the sign of \(q\) and whether \(B\) increases or decreases. --- ## Final Answer \[ \boxed{v_2 = v_1 - \frac{q a}{2m} (B_2 - B_1)} \] Where: - \(v_1\) = initial bead velocity, - \(q\) = bead's charge, - \(a\) = loop radius, - \(m\) = bead's mass, - \(B_1\), \(B_2\) = initial and final magnetic fields (proportional to \(I_1, I_2\)). **Summary:** Yes, the bead's velocity changes as a result of the changing solenoid current due to the induced electric field (Faraday's Law). The exact change is given by the formula above.

## Step 1: Understanding the Physical Situation The solenoid generates a magnetic field \( B \) along its axis, affecting the circular loop \( L \) that is coaxial with it. A charged bead \( A \), with charge \( q \) and mass \( m \), slides along the loop without friction. At time \( t = 0 \), the solenoid has a current \( I_1 \) and a magnetic field \( B_1 \), while the bead has an initial velocity \( v_1 \). When the current changes to \( I_2 \) at time \( t = \tau \), the magnetic field changes from \( B_1 \) to \( B_2 \). **Explanation:** The change in the magnetic field through the loop induces an electromotive force (EMF) in the loop, according to Faraday's Law. This EMF generates an electric field that exerts a force on the charged bead, which can change its velocity. --- ## Step 2: Applying Faraday's Law and Newton's 2nd Law **Faraday’s Law:** The induced EMF around the loop is given by \[ \mathcal{E} = -\frac{d\Phi_B}{dt} \] where \(\Phi_B = B \cdot A = B \cdot \pi a^2\) (area of the loop). Thus, the induced EMF is: \[ \mathcal{E} = -\pi a^2 \frac{dB}{dt} \] This EMF creates a tangential electric field \( E \) around the loop: \[ E = \frac{\mathcal{E}}{2\pi a} = -\frac{a}{2} \frac{dB}{dt} \] The force \( F \) acting on the bead is: \[ F = qE = q \left(-\frac{a}{2} \frac{dB}{dt}\right) \] Using Newton’s second law, we have: \[ F = m \frac{dv}{dt} \] Setting the forces equal gives: \[ m \frac{dv}{dt} = -\frac{q a}{2} \frac{dB}{dt} \] Integrating from \( v_1 \) to \( v_2 \) over time \( \tau \): \[ \int_{v_1}^{v_2} m\,dv = -\frac{q a}{2} \int_{B_1}^{B_2} dB \] This results in: \[ m(v_2 - v_1) = -\frac{q a}{2}(B_2 - B_1) \] Rearranging gives: \[ v_2 = v_1 - \frac{q a}{2m} (B_2 - B_1) \] **Explanation:** The bead's velocity changes due to the induced electric field resulting from the changing magnetic field. The direction and magnitude of this change depend on the charge \( q \) of the bead and the nature of the change in the magnetic field. --- ## Final Answer \[ \boxed{v_2 = v_1 - \frac{q a}{2m} (B_2 - B_1)} \] Where: - \( v_1 \) = initial velocity of the bead, - \( q \) = charge of the bead, - \( a \) = radius of the loop, - \( m \) = mass of the bead, - \( B_1 \), \( B_2 \) = initial and final magnetic fields (related to \( I_1 \) and \( I_2 \)). **Summary:** Yes, the bead's velocity changes due to the induced electric field from the changing solenoid current. The change in velocity is given by the formula above.