The bent wire shown in Fig. 28 42 lies in a uniform magnctic ficld. Each straight section is 2.0

The bent wire shown in Fig. 28 42 lies in a uniform magnctic...

Key Concepts

Lorentz Force

This concept describes the force experienced by a current-carrying conductor when placed in a magnetic field. The force is given by the equation F = I (L × B), where I is the current, L is the length vector of the conductor, and B is the magnetic field. This principle is fundamental in electromagnetism and aids in understanding how electric currents interact with magnetic fields.

Vector Cross Product

The vector cross product is a mathematical operation used to determine a vector that is perpendicular to two given vectors. In the context of the magnetic force on a current-carrying wire, it is used to calculate the direction and magnitude of the force, ensuring that the resulting force is orthogonal to both the current direction and the magnetic field. This concept is critical when working with problems that require force direction determination in three-dimensional space.

Component Resolution and Unit Vectors

When dealing with problems in electromagnetism, especially those involving angles, it is essential to resolve vectors into their components using unit vectors (such as î, ?, and k?). This process simplifies the calculation of the net force by allowing each component to be independently considered and then recombined. Understanding how to break down vectors based on given angles and directions is key to solving problems involving forces on wires.

Superposition Principle in Electromagnetism

This principle states that when multiple magnetic force contributions act on different segments of a wire, the total net force is the vector sum of these individual forces. In the context of a bent wire, forces on different segments (each possibly oriented in different directions) must be calculated separately and then summed to yield the overall force acting on the wire. It underscores the importance of considering both magnitude and direction when combining forces.

Transcript

00:01 So here is the wind wire here that is shown in this figure current i goes along this and current i goes along this we have two cases here it a point b is four in g direction and for this the b is in x xxit direction that is i had direction right okay so the length are given the current is given the angle theta is also given so let us move to the solution so we are asked to calculate the net magnetic force on the wire right so let us do this turn by turn for this wire let us consider a point here okay so let me use green here so for this for this point magnetic field is along k so it is it lies vertically apart right vertically apart is b and current goes along this so if you use your right hand rule then b is vertically upward right v is vertically apart and current is in direction so a direction perpendicular to both of this is this one right so this is force on this wire and again for this wire if you again use the right hand rule for any point analog us to this point here so for this point the force will be along this this two are symmetric okay to the figure show that exactly now this force has two components one alone one along this one along this now due to the geometry of this figure you can say that this angle is theta so this component of force will be f sine theta and this component of forces f cost here similarly for this point and this force f this will be f cost theta and here this will be f sine theta so for every such points in these two wires this forces on this direction minus x and plus x directions cancel out each other and the forces on the minus of y axis add up right so the total force acting on this wires fnit can be written as twice of f cost theta right now in which direction downwards so minus j hat direction right now what is the force on one wire so force on one wire is b magnet equal i l sine 5 where 5 is the angle between length and b or the direction of current and b and then let me continue this equation cost theta minus j now what is b is given in the question four times magnitude of b is 4 magnitude of current is 2 ampers length is 2 meters and what is the angle between b and l so b is particularly upwards and length is on x y plane so 90 degrees so sine 90 is 1 i don't need to write anything and cost of theta is given 60 degree so cost 60 is 1 by 2 photos are 80 to the 16 so half net is equal to minus 16 in direction of this is the net force all right so we are done with this part of the question now let's move on to the second one it says now the magnetic field is alone x so this is the direction of b now look at the figure here b and i both lie in x y plane right both lie in xy pain so the cross product of i and b when we calculate that force that is f is equal to i l cross b if you calculate this force then the force should lie on the pain on the direction perpendicular to both b and i therefore it lies along j along g now for the first wire for the first wire if you use the right hand rule then f1 lies force lies alone minus k it is downwards into the paper or into the screen whereas for the second wire applies upwards by using right hand rule that is in k hat direct so for every points in this wire there is a point with has force in opposite direction right so all the forces on these wires so force on this point and force on this point are equal and opposite force on this point and this point are equal and opposite so every forces at every points cancel out each other and the net force becomes zero so the net force on the wire will be zero when the magnetic field is along x -axis so we are done with part of the other question as well...

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