Problem 33 - Physics

Question

A copper rod of length \(0.85 \mathrm{~m}\) is lying on a frictionless table (see the drawing). Each end of the rod is attached to a fixed wire by an unstretched spring that has a spring constant of \(k=75 \mathrm{~N} / \mathrm{m}\). A magnetic field with a strength of \(0.16 \mathrm{~T}\) is oriented perpendicular to the surface of the table. (a) What must be the direction of the current in the copper rod that causes the springs to stretch? (b) If the current is \(12 \mathrm{~A}\), by how much does each spring stretch?

Step-by-Step Solution

Answer

(a) Current flows from left to right; (b) Each spring stretches by 0.022 m.

Step 1: Identify the Physics Principles

For part (a), we need to determine the direction of the current that causes a magnetic force to oppose the force of the springs. We apply the right-hand rule for the magnetic force on a current-carrying rod. For part (b), we use Hooke's Law and the formula for magnetic force to find the displacement.

Step 2: Apply the Right-Hand Rule for Part (a)

Use the right-hand rule: point your fingers in the direction of the current and curl them towards the magnetic field direction (into the table). Your thumb will point in the direction of the force exerted on the rod. In order for the springs to stretch, this magnetic force direction must be along the length of the rod pointing away from its center.

Step 3: Determine the Direction of Current for Part (a)

To stretch the springs, the magnetic force needs to act outward on both ends of the rod. If the magnetic field is directed into the table, then by using the right-hand rule, the current should flow from left to right along the rod to create such a force.

Step 4: Calculate Magnetic Force for Part (b)

The magnetic force on the rod can be calculated using the formula: \[ F_B = I L B \]where \( I = 12 \text{ A} \), \( L = 0.85 \text{ m} \), and \( B = 0.16 \text{ T} \). Substituting the values:\[ F_B = 12 \times 0.85 \times 0.16 = 1.632 \text{ N} \]

Step 5: Apply Hooke's Law for Part (b)

The force exerted by each spring is given by Hooke’s Law: \[ F_s = k imes x \]where \( F_s = 1.632 \text{ N} \) is the force applied to each spring to cause stretching (since the rod is in equilibrium, the magnetic force on the rod is balanced by the spring forces), \( k = 75 \text{ N/m} \). Solve for \( x \):\[ x = \frac{F_s}{k} = \frac{1.632}{75} = 0.02176 \text{ m} \approx 0.022 \text{ m} \]

Step 6: Interpretation of Results

For the current to cause the springs to stretch, it should flow from left to right along the rod. Each spring stretches by approximately \(0.022 \, \text{m}\).

Key Concepts

Right-Hand Rule

The right-hand rule is a simple yet powerful tool used in physics to determine the direction of the magnetic force acting on a current-carrying wire. Visualize your right hand to apply this rule:

• Point your thumb in the direction of the current.
• Let your fingers follow the direction of the magnetic field lines.
• Your palm then faces in the direction of the force exerted on the wire.

The rule helps you understand how currents interact with magnetic fields, generating forces that can cause a rod, like the copper rod in our exercise, to move in specific directions. It's crucial to visualize the field direction properly—for the exercise, the magnetic field is directed perpendicularly into the table.

Hooke's Law

Hooke's Law is a fundamental principle relating the force needed to extend a spring to the distance it is extended. The law can be expressed as: \( F_s = k \times x \). Where:

• \( F_s \) denotes the force exerted by the spring.
• \( k \) is the spring constant, indicating the stiffness of the spring.
• \( x \) is the displacement, or the amount by which the spring is stretched or compressed from its original length.

In our problem, Hooke's Law helps us determine the stretch of the springs induced by the magnetic force. When we calculate the force exerted by the current in the magnetic field, this force is balanced by the spring forces, allowing us to find how far the springs extend.

Magnetic Field

A magnetic field is a region around a magnetic object where magnetic forces are experienced. These fields are represented by vectors that begin at the north pole of a magnet and move toward the south pole. Here are some essential characteristics to note about magnetic fields:

• The direction of the field is indicated by field lines, showing how a magnetic north pole would experience force.
• Field strength is measured in teslas (T), with the magnetic field in our problem being \( 0.16 \, \text{T} \).
• Field patterns can be visualized using lines which show movement, such as the field going into or out of a surface.

In our scenario, the magnetic field is perpendicular to the table and impacts how the copper rod experiences forces, interacting with the current to stretch the springs attached to the rod.

Spring Constant

The spring constant \( k \) is a critical factor in defining how resistant a spring is to being compressed or stretched. It is a measure of stiffness, indicating the force required to elongate or compress the spring by a unit of length. Here are a few key points about the spring constant:

• Measured in newtons per meter (N/m), it describes the rigidity of the spring. In our exercise, \( k = 75 \, \text{N/m} \).
• A higher spring constant means a stiffer spring, requiring more force to achieve the same amount of stretch as a less stiff spring.
• The spring constant plays a central role in Hooke's Law, influencing how much a spring stretches or compresses under a given force.

Understanding the spring constant allows us to solve how much each spring in our exercise stretches when subjected to the magnetic force.