Two long, parallel wires separated by a distance \(r\) carry equal currents \(I\) in opposite directions, as shown. The direction of the field caused by the top wire at the position of the bottom wire and the direction of the force exerted by the top wire on the bottom wire are (A) \(B\) into the page; \(F\) down (B) \(B\) up; \(F\) into the page (C) \(B\) into the page; \(F\) up (D) \(B\) out of the page; \(F\) down (E) \(B\) down; \(F\) out of the page

Short Answer

Expert verified
Answer: The magnetic field (B) is into the page, and the force (F) is downward.

Step by step solution

01

Understanding the right-hand rule

The right-hand rule is a useful mnemonic tool to determine the direction of the magnetic field generated by a current-carrying wire. We point our thumb in the direction of the current, and our fingers curl in the direction of the magnetic field.
02

Determine the magnetic field at the bottom wire's location

Using the right-hand rule, we can determine the direction of the magnetic field generated by the top wire at the position of the bottom wire. The top wire has a current flowing to the left, so pointing our thumb to the left (representing the current direction) shows that the magnetic field at the bottom wire's location curls into the page.
03

Understand the Lorentz force and its direction

The Lorentz force is the force on a charged particle moving within a magnetic field. In this case, the charged particles are the current flowing through the bottom wire. The direction of the Lorentz force (\(F\)) can be found using another variant of the right-hand rule: point index finger in the direction of current (I), middle finger in the direction of magnetic field (B), and thumb will point in the direction of force (F).
04

Calculate the force direction

Using the Lorentz force right-hand rule, point the index finger in the direction of the current in the bottom wire (to the right), and the middle finger in the direction of the magnetic field found in step 2 (into the page). The thumb now points downward, indicating that the direction of the force exerted on the bottom wire by the top wire is downward. Therefore, the answer is (A) \(B\) into the page; \(F\) down.

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