loop in between two currents

loop in between two currents - 2 c I ( t ) Z s Z 1 2 s 1...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Square loop of wire in between two long current-carrying wires March 27, 2008 Problem: Two long straight wires both initially carrying a steady current of I 0 are separated by distance d . Half way between the wires there is a square loop of wire with sides of length s and resistance R . At t = 0 the current in the wires begins to decrease as I ( t ) = I 0 e - t τ (a.) Find the total amount of charge that flows through any point in the wire as t → ∞ . (b.) Find how much work was done on the loop of wire. Solution: For this problem we will define the polar radius to be r, r 2 = x 2 + y 2 , and that the long wires are along the z-axis and located at x = ± d 2 . (a.) First describe the total magnetic field from the long wires B 1 ( r ,t ) = 2 I ( t ) c 1 r - d 2 δ ( φ ) ˆ φ B 2 ( r ,t ) = 2 I ( t ) c 1 r - d 2 δ ( φ - π ) ˆ φ B ( r ,t ) = B 1 ( r ,t ) + B 2 ( r ,t ) Now we will first find the flux through the square loop as a function of time. Φ( t ) = Z S B · ˆ nda = Z S B ˆ φ · ˆ nda Φ( t ) = Z s 0 Z 1 2 s 0 B ( φ = 0) ˆ φ · ˆ φdr dz - Z s 0 Z 1 2 s 0 B ( φ = π ) ˆ φ · ( - ˆ φ ) dr dz Φ( t ) =
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2 c I ( t ) Z s Z 1 2 s 1 r-d 2 dr dz + 2 c I ( t ) Z s Z 1 2 s 1 r-d 2 dr dz = 4 c I ( t ) Z s dz Z 1 2 s 1 r-d 2 dr ( t ) = 4 s c I ( t ) ln 1 2 ( s-d ) -ln -d s ( t ) = 4 s c I e-t ln d-s d Using the magnetic ux through the loop we can nd the EMF on the loop. V ( t ) =-d dt = 4 s c I 1 e-t ln d-s d 1 Now to nd the total charge that will ow through any point in the loop we simply integrate the current in the loop over all time. Q = Z I loop dt = 1 R Z V ( t ) dt =-1 R 4 s c I ln d-s d Z -1 e-t dt Q = 4 s Rc I ln d-s d (b.) To nd the work done on the loop we do the following. W tot = Z I 2 ( t ) R dt = Z 1 R V 2 ( t ) dt 2...
View Full Document

Page1 / 2

loop in between two currents - 2 c I ( t ) Z s Z 1 2 s 1...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online