Let's solve the problem step by step using the **right-hand rule** for the magnetic force on a current-carrying wire: ### Magnetic Force on Current-Carrying Wire The force on a wire of length **L** carrying current **I** in a magnetic field **B** is: \[ \vec{F} = I(\vec{L} \times \vec{B}) \] Use the **right-hand rule**: Point your fingers in the direction of current (\(\vec{I}\)), curl them toward the direction of the magnetic field (\(\vec{B}\)), and your thumb points in the direction of the force. --- ## **a) Magnetic Field Into the Page (\(\vec{B}\) into page, shown as "×")** ### 1. Left wire (Current up) - **Current (\(\vec{I}\))**: Up (along y-axis) - **Magnetic Field (\(\vec{B}\))**: Into the page (−z) - **Right-hand rule**: Fingers up, curl into the page - **Thumb points**: Left (−x direction) - **Force**: **Left** ### 2. Middle wire (Current right) - **Current (\(\vec{I}\))**: Right (along x-axis) - **Magnetic Field (\(\vec{B}\))**: Into the page (−z) - **Right-hand rule**: Fingers right, curl into the page - **Thumb points**: Down (−y direction) - **Force**: **Down** ### 3. Bottom wire (Current out of page, "dot") - **Current (\(\vec{I}\))**: Out of the page (+z) - **Magnetic Field (\(\vec{B}\))**: Into the page (−z) - **Right-hand rule**: Fingers out of page, curl into the page - **Thumb points**: There is **no force** (since current and field are parallel/antiparallel, cross product is zero). - **Force**: **No force** --- ## **b) Magnetic Field Downward (\(\vec{B}\) down, shown as arrows down)** ### 1. Left wire (Current into page, "×") - **Current (\(\vec{I}\))**: Into page (−z) - **Magnetic Field (\(\vec{B}\))**: Downward (−y) - **Right-hand rule**: Fingers into page, curl down - **Thumb points**: Right (+x) - **Force**: **Right** ### 2. Middle wire (Current out of page, "dot") - **Current (\(\vec{I}\))**: Out of page (+z) - **Magnetic Field (\(\vec{B}\))**: Downward (−y) - **Right-hand rule**: Fingers out of page, curl down - **Thumb points**: Left (−x) - **Force**: **Left** ### 3. Right wire (Current right, +x) - **Current (\(\vec{I}\))**: Right (+x) - **Magnetic Field (\(\vec{B}\))**: Downward (−y) - **Right-hand rule**: Fingers right, curl down - **Thumb points**: Into page (−z) - **Force**: **Into the page** --- ## **Summary Table** | Diagram | Wire | Current Direction | Magnetic Field | Force Direction | |---------|--------------|---------------------|-------------------|---------------------| | (a) | Left | Up (+y) | Into page (−z) | Left (−x) | | (a) | Middle | Right (+x) | Into page (−z) | Down (−y) | | (a) | Bottom | Out of page (+z) | Into page (−z) | **No force** | | (b) | Left | Into page (−z) | Down (−y) | Right (+x) | | (b) | Middle | Out of page (+z) | Down (−y) | Left (−x) | | (b) | Right | Right (+x) | Down (−y) | Into page (−z) | --- **Final Answers (with explanations above):** ### (a) - Left wire: **Force to the left** - Middle wire: **Force down** - Bottom wire: **No force** ### (b) - Left wire: **Force to the right** - Middle wire: **Force to the left** - Right wire: **Force into the page** Let me know if you'd like the forces drawn on the diagrams!

The magnetic force on a current-carrying wire is determined by the right-hand rule, where fingers point in the current direction and curl toward the magnetic field; the thumb then indicates the force direction. When the magnetic field is into or out of the page, the force depends on the current direction, resulting in forces perpendicular to both current and field. If the current is parallel to the magnetic field, the force is zero because the cross product is zero. Thus, the force direction varies based on current orientation and magnetic field direction, following vector cross product principles.