A copper rod of length 0.85 m is lying on a frictionless table (see the drawing). Each end of th

A copper rod of length 0.85 m is lying on a frictionless...

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?

Key Concepts

Magnetic Force on a Current-Carrying Conductor

This concept explains that a current-carrying conductor placed in a magnetic field experiences a force given by the equation F = I L × B. It is fundamental in understanding how the magnetic field interacts with the current within the conductor to provide the force that ultimately affects the mechanical equilibrium of the system.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the force acting on a current-carrying conductor in a magnetic field. By aligning the thumb, forefinger, and middle finger appropriately, one can deduce the orientation of the force relative to the direction of the current and the magnetic field.

Hooke’s Law

Hooke’s Law relates the force exerted by a spring to its extension or compression, mathematically expressed as F = k?x, where k is the spring constant and ?x is the displacement. This principle is key for analyzing how much a spring stretches when subjected to the force generated by the magnetic interaction.

Mechanical Equilibrium

Mechanical equilibrium in this context refers to the state where the magnetic force acting on the rod is exactly balanced by the restoring force of the springs. This balance allows for the calculation of the spring extension by equating the force from the magnetic field with the spring force as given by Hooke's Law.

Transcript

00:01 Okay, so i have a little setup here, a little drawing of the setup of the situation.

00:09 So we have a table, even if it's a poorly draw table, it is still a table.

00:13 We have this copper rod here attached by springs to wires.

00:21 So this is all laid out flat on the table.

00:26 That was the only reason i drew the table here for you to see that it's laid flat on it.

00:33 Since this is a three -dimensional problem that asks for direction, it can be helpful to get a good visual of what's happening.

00:39 So we have this magnetic field that i have in blue that's coming straight up out of the table.

00:46 So it's perpendicular to the rod, the wires, and the springs.

00:51 Okay.

00:52 Now the first part is the direction part.

00:55 We want to know what direction we need to have current in the red wire here for the springs to be stretched.

01:03 And to be stretched means that the wire, or the copper rod needs to move down.

01:12 That will stretch.

01:13 Otherwise, it would be compressing the springs.

01:15 We want to find what direction the current needs to be in for it to stretch the spring.

01:20 So the force would have to go down to pull it away from the wire.

01:24 Now, we have to use the right -hand rule to find the direction here.

01:28 So the right -hand rule in this case, you point your fingers in the direction of the magnetic field.

01:34 So in this case, it will be up out of the page or up off the table.

01:40 And your palm, or you could point your middle finger down, so it's perpendicular to your pointer finger, whichever method you are familiar with.

01:50 If you point your fingers up, your palm should be facing straight back towards you, which is the force direction, which is this direction.

01:57 We want the copper rod to go in.

02:00 And then your thumb, for both of those directions, your thumb will point the direction that the current would need to flow to give you those directions.

02:09 So because the force has to go down and we know the magnetic field is up, our thumb should be pointing in the right direction.

02:18 So the current has to be flowing from left to right for it to have that force.

02:24 So for part a, the current has to flow to the right, from left to right.

02:44 Okay, now for part b, we are asked to find how far these springs compress, or how far they stretch.

02:56 So now we need to pull back out the equation for the force from spring, deep in our memories.

03:04 If you don't remember it, i'll write it here.

03:06 We have the force for a spring is equal to negative k, which is the spring constant, times the distance.

03:18 This negative sign we don't need to be concerned with.

03:21 All it means is that it's the distance from equilibrium, either stretched or pulled.

03:28 If you leave a spring to its own devices, it will go to its equilibrium position.

03:32 If you stretch it or pull it, it will try to go back to its equilibrium.

03:36 So that means this k is always opposite whatever direction it's being pulled in or pushed.

03:43 It won't really factor into this problem here.

03:47 We are given k...

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