A device called a railgun uses the magnetic force on...

A device called a railgun uses the magnetic force on currents to launch projectiles at very high speeds. An idealized model of a railgun is illustrated in Figure P24.48. A 1.2 V power supply is connected to two conducting rails. A segment of copper wire, in a region of uniform magnetic field, slides freely on the rails. The wire has a $0.85 \mathrm{m} \Omega$ resistance and a mass of $5.0 \mathrm{g}$. Ignore the resistance of the rails. When the power supply is switched on, a. What is the current? b. What are the magnitude and direction of the force on the wire? c. What will be the wire's speed after it has slid a distance of $1.0 \mathrm{mm} ?$

A device called a railgun uses the magnetic force on currents to launch projectiles at very high speeds. An idealized model of a railgun is illustrated in Figure P24.48. A 1.2 V power supply is connected to two conducting rails. A segment of copper wire, in a region of uniform magnetic field, slides freely on the rails. The wire has a $0.85 \mathrm{m} \Omega$ resistance and a mass of $5.0 \mathrm{g}$. Ignore the resistance of the rails. When the power supply is switched on,
a. What is the current?
b. What are the magnitude and direction of the force on the wire?
c. What will be the wire's speed after it has slid a distance of $1.0 \mathrm{mm} ?$

Key Concepts

Ohm's Law

Ohm’s Law is a fundamental principle in electrical circuits, which relates the voltage (V), current (I), and resistance (R) through the equation I = V/R. This law allows for the calculation of the current flowing through a conductor when the voltage across it and its resistance are known, and is a key concept for analyzing circuits, such as in devices like railguns where a voltage source drives a current through a resistive element.

Lorentz Force on a Current-Carrying Conductor

The Lorentz force describes the force exerted on a current-carrying conductor when it is placed in a magnetic field. The force is given by F = I L × B, where I is the current, L is the length vector (direction of current), and B is the magnetic field. This concept is central to the operation of railguns, as it explains how a force is generated on the wire, producing acceleration and eventually launching a projectile.

Newton's Second Law of Motion

Newton's Second Law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration (F = m·a). This law is used to determine how the force generated by the Lorentz force affects the acceleration of the railgun's moving part. Understanding this connection between force, mass, and acceleration is essential for predicting the motion of objects subjected to a net force.

Uniformly Accelerated Motion

Uniformly accelerated motion deals with the kinematics of objects moving under constant acceleration. Equations from this domain, such as v² = u² + 2a·s or v = u + at, allow for the calculation of the final velocity or displacement of an object when its acceleration and either time or distance traveled are known. This is crucial when determining the speed of the projectile in a railgun after moving over a specified distance.

Transcript

00:02 Hi, in the given problem in the rail gun experiment or we can say a device known as rail gun, there are two rails over which there is a rod, the conducting rod.

00:21 These two rails are joined with the help of a battery having an emf of 1 .2 volts.

00:31 So this is epsilon is equal to 1 .2 volt.

00:35 The magnetic field in this region is vertically downward, inward into the plane of paper, not vertically downward.

00:47 It is inward into the plane of paper.

00:50 The gap between the rails, or we can say the length of the rod, which is moving over the rails, is given as 15 centimeter, means here this l, is 15.

01:03 Centimeter or we can say this is 0 .15 meter.

01:08 The resistance of the circuit is 0 .85 milli -oam or we can say this is 0 .85 into 10 dash to power minus 3 ome.

01:22 Mass of this rod is given as 5 .0 gram and the magnetic field in which this rod will be moving is having a magnitude of 0 .80 tesla.

01:36 So in the first part of the problem, we have to find just the magnitude of current i passing through the rod.

01:45 That is given by oms law simply i is equal to emf applied by r.

01:51 So this is 1 .2 volt divided by 0 .85 into 10ish per minus 3 ome.

02:00 So finally this current passing through the rod comes out to be 1 ,411 .8 ampere.

02:09 And this is the answer for the first part of the problem now.

02:14 In the second part of the problem, we have to find the force, the magnetic force acting on this conducting rod, which is given by the expression b i l, where b is 0 .80 tesla multiplied by current, which we just found.

02:32 1411 .8 ampere multiplied by the length of the rod means the gap between the two rails, which is 0 .15 meter...

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