Let's break the problem into two parts as requested, with detailed explanations: --- ### **Step 1: Find the Magnetic Flux Through the Loop** #### **Explanation** A long straight wire carries a current \( I \) upward. The magnetic field at a distance \( r \) from a long straight wire is given by: \[ B = \frac{\mu_0 I}{2 \pi r} \] where \( \mu_0 \) is the permeability of free space. For a rectangular loop of width \( w \) and length \( L \), placed such that its near side is at distance \( a \) and its far side at distance \( a + w \): - The magnetic field varies with \( r \), so we need to integrate over the width of the loop. The infinitesimal flux through a strip of width \( dr \) at distance \( r \) from the wire: \[ d\Phi = B(r) \cdot L \cdot dr = \frac{\mu_0 I}{2\pi r} L dr \] Integrate \( r \) from \( a \) to \( a + w \): \[ \Phi = \int_a^{a+w} \frac{\mu_0 I}{2\pi r} L dr = \frac{\mu_0 I L}{2\pi} \int_a^{a+w} \frac{dr}{r} \] \[ \Phi = \frac{\mu_0 I L}{2\pi} \left[ \ln(r) \right]_a^{a+w} \] \[ \Phi = \frac{\mu_0 I L}{2\pi} \ln\left( \frac{a + w}{a} \right) \] **Boxed Final Answer for Step 1:** \[ \boxed{ \Phi = \frac{\mu_0 I L}{2\pi} \ln\left( \frac{a + w}{a} \right) } \] --- ### **Step 2: Find the Induced EMF in the Loop** #### **Explanation** The induced EMF (\( \mathcal{E} \)) is given by Faraday's Law: \[ \mathcal{E} = - \frac{d\Phi}{dt} \] Since \( L, a, w \), and \( \mu_0 \) are constants, only \( I \) changes with time: \[ \mathcal{E} = - \frac{d}{dt} \left( \frac{\mu_0 I L}{2\pi} \ln \left( \frac{a + w}{a} \right) \right) \] \[ \mathcal{E} = - \frac{\mu_0 L}{2\pi} \ln \left( \frac{a + w}{a} \right) \frac{dI}{dt} \] Given \(\frac{dI}{dt} = k\) (as stated in the problem that the current increases steadily at a rate \( k \)), the magnitude of the induced EMF is: \[ \boxed{ \mathcal{E} = \frac{\mu_0 L}{2\pi} \ln \left( \frac{a + w}{a} \right) k } \] --- #### **Summary of Steps and Answers:** **(a) Magnetic flux through the loop:** \[ \Phi = \frac{\mu_0 I L}{2\pi} \ln\left( \frac{a + w}{a} \right) \] **(b) Induced emf in the loop:** \[ \mathcal{E} = \frac{\mu_0 L}{2\pi} \ln\left( \frac{a + w}{a} \right) k \] Let me know if you'd like a step-by-step diagram or further clarification!

Let's analyze the problem step by step with detailed explanations. --- ### **Step 1: Calculate the Magnetic Flux Through the Loop** #### **Understanding the problem** - A long straight wire carries a current \( I \) directed upward. - The magnetic field \( B \) created by this current at a distance \( r \) from the wire is given by: \[ B(r) = \frac{\mu_0 I}{2\pi r} \] - The loop is a rectangle positioned at distances \( a \) (near side) and \( a + w \) (far side) from the wire, with length \( L \) along the wire. - To find the total magnetic flux \( \Phi \) through the loop, we integrate \( B(r) \) over the width \( w \): \[ \Phi = \int_{r=a}^{r=a+w} B(r) \times \text{area element} \] Since the magnetic field is perpendicular to the plane of the loop (assuming the loop is in a plane perpendicular to the wire), the flux through a strip at distance \( r \) is: \[ d\Phi = B(r) \times L \, dr \] where \( dr \) is an infinitesimal width element at distance \( r \). #### **Calculating the magnetic flux** \[ \Phi = \int_{a}^{a+w} \frac{\mu_0 I}{2\pi r} \times L \, dr = \frac{\mu_0 I L}{2\pi} \int_{a}^{a+w} \frac{1}{r} dr \] \[ \Phi = \frac{\mu_0 I L}{2\pi} \left[ \ln r \right]_a^{a+w} = \frac{\mu_0 I L}{2\pi} \ln \left( \frac{a + w}{a} \right) \] --- ### **Step 2: Find the Induced EMF in the Loop** #### **Understanding Faraday's Law** Faraday's law states: \[ \mathcal{E} = - \frac{d\Phi}{dt} \] - The magnetic flux \( \Phi \) depends on the current \( I(t) \). If \( I \) varies with time, so does \( \Phi \). - Assuming the current changes at a constant rate: \[ \frac{dI}{dt} = k \] - The flux \( \Phi \) then varies with \( I(t) \): \[ \Phi(t) = \frac{\mu_0 L}{2\pi} \ln \left( \frac{a + w}{a} \right) I(t) \] - Taking the derivative with respect to time: \[ \frac{d\Phi}{dt} = \frac{\mu_0 L}{2\pi} \ln \left( \frac{a + w}{a} \right) \frac{dI}{dt} = \frac{\mu_0 L}{2\pi} \ln \left( \frac{a + w}{a} \right) k \] - The induced EMF (magnitude) is: \[ \boxed{ \mathcal{E} = \frac{\mu_0 L}{2\pi} \ln \left( \frac{a + w}{a} \right) k } \] --- ### **Final answers:** - **Magnetic flux through the loop:** \[ \boxed{ \Phi = \frac{\mu_0 I L}{2\pi} \ln \left( \frac{a + w}{a} \right) } \] - **Induced EMF in the loop:** \[ \boxed{ \mathcal{E} = \frac{\mu_0 L}{2\pi} \ln \left( \frac{a + w}{a} \right) k } \] --- If you'd like, I can also include a diagram to visualize the setup!