Q14P Suppose the current density chan... [FREE SOLUTION]

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Introduction to Electrodynamics

David J. Griffiths

Physics

4th edition Edition 路 613 pages

ISBN: 9780321856562

Chapter 10: Q14P (page 450)

Suppose the current density changes slowly enough that we can (to good approximation) ignore all higher derivatives in the Taylor expansion
$J (t_{r}) = J (t) + (t_{r} - t) J (t) + 鈥�$
(for clarity, I suppress the r-dependence, which is not at issue). Show that a fortuitous cancellation in Eq. 10.38 yields
$B (r, t) = \frac{渭_{0}}{4 蟺} 鈭� \frac{J (r', t) 脳 \hat{r}}{r^{2}} d b'$ .
That is: the Biot-Savart law holds, with J evaluated at the non-retarded time. This means that the quasistatic approximation is actually much better than we had any right to expect: the two errors involved (neglecting retardation and dropping the second term in Eq. 10.38 ) cancel one another, to first order.

Short Answer

Expert verified

The fortuitous cancellation in Eq. 10.38 yields $B (r, t) = \frac{渭_{0}}{4 蟺} 鈭� [\frac{J (r', t) 脳 \hat{r}}{r^{2}}] d b'$ .

Step by step solution

Expression for the current density and magnetic field strength.

Write the expression for the current density.

$J (t_{r}) = J (t) + (t_{r} - t) + J (t) + 鈥�$

Here, J is the current density and $t_{r}$ is the retarded time.

Write the expression for the magnetic field strength.

$B (r, t) = \frac{渭_{0}}{4 蟺} 鈭� [\frac{J (r', t_{r})}{r^{2}} + \frac{J (r, t_{r})}{c r}] 脳 \hat{r} d b'$ 鈥︹€� (1)

Here, B is the magnetic field, $渭_{0}$ is the permeability of free space and c is the speed of light.

Prove B(r,t)=μ04π∫[Jr',t×r^r2]db' :

Write the expression for the retarded time.

$t_{r} = t - \frac{r}{c}$

Substitute the value of $t_{r}$ in equation (1).

$B (r, t) = \frac{渭_{0}}{4 蟺} 鈭� \{[\frac{J (r', t)}{r^{2}} + \frac{(t_{r} - t) J (t)}{r^{2}}] + [\frac{J (r', t_{r})}{c r}]\} 脳 \hat{r} d b' B (r, t) = \frac{渭_{0}}{4 蟺} 鈭� \frac{1}{r^{2}} [J (r', t) + (t_{r} - t) J (t) + (\frac{r}{c}) J (r', t_{r})] 脳 \hat{r} d b' B (r, t) = \frac{渭_{0}}{4 蟺} 鈭� \frac{1}{r^{2}} [J (r', t) - \frac{r}{c} J (r', t) + \frac{r}{c} J (r', t_{r 1})] 脳 \hat{r} d b' B (r, t) = \frac{渭_{0}}{4 蟺} 鈭� [\frac{J (r', t) 脳 \hat{r}}{r^{2}}] d b'$

Therefore, the fortuitous cancellation in Eq. 10.38 yields

$B (r, t) = \frac{渭_{0}}{4 蟺} 鈭� [\frac{J (r', t) 脳 \hat{r}}{r^{2}}] d b'$ .

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Short Answer

Step by step solution

Expression for the current density and magnetic field strength.

Prove B(r,t)=μ04π∫[Jr',t×r^r2]db' :

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