**Problem 3:**A loop of wire in the shape of a rectangle with width \( w \) and length \( L \) and a long, straight wire carrying a current \( I \) lie on a tabletop (see diagram).**a)** Suppose the current is changing with time according to \( I = a + bt \) where \( a \) and \( b \) are constants. Determine the emf that is induced in the loop if \( b = 10 \, \text{A/s} \); \( h = 1 \, \text{cm} \); \( w = 10 \, \text{cm} \); and \( L = 1.0 \, \text{m} \).**b)** What is the direction of the induced current in the rectangle?(Note: The problem mentions a diagram which is not provided here, so the setup and any necessary visual details are not available for description.)

The change in magnetic flux linked with a closed circuit with time will always produce an electric current in the circuit. This phenomenon is known as electromagnetic induction. The value and direction of emf is given by Faraday's law of electromagnetic induction. There are two points in this law The induced emf in a circuit is proportional to the rate of change of magnetic flux linked with the circuit. The direction of the induced emf is such that it always tries to oppose the cause by which it is produced. The second point is known as Lenz's Law. Mathematically Farady's law can be expressed as E=-dϕdt Here E is the EMF induced and ϕ is the magnetic flux. The negative sign is has been included for Lenz's law. If S is an open surface in a magnetic field B→ then the magnetic flux through the surface S is ϕ=∫SB→.dS→ This is known as the surface integral. Thus the emf due to this is E=-ddt∫SB→.dS→ Since the diagram is missing from the question, I will assume the following configuration for my calculation. Since it has been mentioned that the wire is long we can consider it to be an infinitely long wire. This wire is producing a time-dependent magnetic field. at a distance r the magnetic field due to this wire is B=μ0I2πr Flux due to the elementary area dS=L dr is dϕ=B dS=μ0IL2πrdr Therefore flux due to entire rectangle ϕ=μ0IL2π∫hh+wdrr=μ0IL2πlnh+wh Therefore the induced emf E=-dϕdt=-μ0L2πlnh+whdIdt=--μ0L2πlnh+whb Therefore the emf induced is E=4π×10-7×12π×ln0.001+0.010.01×10=1.90×10-7V=0.19μV The direction of the magnetic lines of force is into the plane of the screen. Therefore the direction of induced emf is clockwise as seen from above Answer: a. Emf induced in the loop is E=0.19 μV b. Direction of induced emf is clockwise if viewed from above.

Science

Advanced Physics

a) Suppose the current is changing with time according to I = a + bt whe constants. Determine the emf that is induced in the loop if b = 10 A/ and L= 1.0 m. b) What is the direction of the induced current in the rectangle?

a) Suppose the current is changing with time according to I = a + bt whe constants. Determine the emf that is induced in the loop if b = 10 A/ and L= 1.0 m. b) What is the direction of the induced current in the rectangle?