loop in between two currents

# loop in between two currents - 2 c I t Z s Z 1 2 s 1 r-d 2...

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

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 ) =

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

View Full Document
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 Φ( t ) = 4 s c I ( t ) ± ln ± 1 2 ( s-d ) ²-ln ±-d s ²² Φ( t ) = 4 s c I e-t τ ln ± d-s d ² Using the magnetic ﬂux through the loop we can ﬁnd the EMF on the loop. V ( t ) =-d Φ dt = 4 s c I ± 1 τ ² e-t τ ln ± d-s d ² 1 Now to ﬁnd the total charge that will ﬂow 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 ²Z ∞-1 τ e-t τ dt Q = 4 s Rc I ln ± d-s d ² (b.) To ﬁnd 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

{[ snackBarMessage ]}

### Page1 / 2

loop in between two currents - 2 c I t Z s Z 1 2 s 1 r-d 2...

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

View Full Document
Ask a homework question - tutors are online