Let's break down each part step by step: --- ### (i) **Sketch a graph of the induced current \(I\) as a function of time \(t\) from \(t = 0\) to \(t = 18\,s\)** #### **Step 1: Understanding the Problem** - We need more information about the changing magnetic flux or the scenario (usually, the problem would state if the field is increasing, decreasing, or reversing over certain intervals). - Since the question is standard, let's assume the typical scenario: - From \( t = 0 \) to \( t = 8\,s \): The magnetic flux through the loop is **increasing**. - From \( t = 8\,s \) to \( t = 12\,s \): The flux is **constant**. - From \( t = 12\,s \) to \( t = 16\,s \): The flux is **decreasing**. - After \( t = 16\,s \): The flux is **constant** again. #### **Step 2: Lenz's Law** - **Increasing flux:** Induced current is in a direction that opposes the increase (CCW or CW depending on the field). - **Constant flux:** No induced current. - **Decreasing flux:** Induced current is in a direction that opposes the decrease (opposite direction to the first interval). #### **Step 3: Drawing the Graph** - **From \(0\) to \(8\,s\):** Induced current is positive (CCW). - **From \(8\) to \(12\,s\):** Induced current is zero (no change in flux). - **From \(12\) to \(16\,s\):** Induced current is negative (CW). - **From \(16\) to \(18\,s\):** Induced current is zero. ##### **Graph:** ``` I (A) | _______ _______ | | | | | | | | |_________| |__________|_________ t(s) 0 8 12 16 18 ``` - From 0 to 8 s, I is **positive** (CCW). - From 8 to 12 s, I is **zero**. - From 12 to 16 s, I is **negative** (CW). - From 16 to 18 s, I is **zero**. --- ### (ii) **For the time interval \(12\,s\) to \(16\,s\), justify the direction of the current in your graph.** #### **Answer:** - **From \(12\,s\) to \(16\,s\), the magnetic flux through the loop is decreasing.** - According to **Lenz's Law**, the induced current flows in a direction that creates a magnetic field to oppose the decrease in flux. - If the original field was increasing into the page (for example), the induced current was CCW from 0 to 8 s. - When the field decreases (12 to 16 s), the induced current must try to keep the field from decreasing—**so it reverses direction (CW)**, which is shown as negative on the graph. --- ### (iii) **Calculate the total energy dissipated in the loop during the first \(8\,s\) shown.** #### **Step 1: Formula for Energy Dissipated** - Energy dissipated in a resistor: \( E = I^2 R t \) - We need to know the value of \(I\) and \(R\). - If the current is constant during the first 8 s (as per the graph), and let's say \(I_0\) is the value. #### **Step 2: If \(I_0\) is not given** - If not given, leave the answer in terms of \(I_0\) and \(R\): \[ E = I_0^2 R \times 8 \] #### **Step 3: If Specific Values are Given** - Plug in those values if specified in the full problem. --- ## **Final Answers** **i. Sketch:** - From \(0\) to \(8\,s\): \(I\) is positive (CCW). - From \(8\) to \(12\,s\): \(I = 0\). - From \(12\) to \(16\,s\): \(I\) is negative (CW). - From \(16\) to \(18\,s\): \(I = 0\). **ii. Justification:** - From \(12\,s\) to \(16\,s\), the magnetic flux is **decreasing**, so by Lenz's Law, the current reverses direction (CW) to oppose the decrease. **iii. Energy dissipated (first \(8\,s\)):** \[ E = I_0^2 R \times 8 \] (where \(I_0\) is the constant induced current and \(R\) is the resistance of the loop). --- If you have the specific values for \(I_0\) and \(R\), I can substitute them into the equation for you!