91Ó°ÊÓ

The current in a long solenoid is increasing linearly with time, so the flux is proportional $t$:.$\mathrm{Ï•}=\mathrm{Î\pm }t$Two voltmeters are connected to diametrically opposite points (A and B), together with resistors ( $R_1$and $R_2$), as shown in Fig. 7.55. What is the reading on each voltmeter? Assume that these are ideal voltmeters that draw negligible current (they have huge internal resistance), and that a voltmeter register -$-{∫}_a^{b}E×dl$between the terminals and through the meter. [Answer: $V_1=\mathrm{Î\pm }{R}_1/(R_1+R_2)$. Notice that $V_1≠V_2$, even though they are connected to the same points]

An infinite wire runs along the z axis; it carries a current I (z) that is a function ofz(but not of t ), and a charge density $\mathit{λ}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ that is a function of t (but not of z ).

(a) By examining the charge flowing into a segment dz in a time dt, show that $\mathit{d}\mathit{λ}\mathbf{/}\mathit{d}\mathit{t}\mathbf{=}\mathbf{-}\mathit{d}\mathit{i}\mathbf{/}\mathit{d}\mathit{z}$. If we stipulate that $\mathit{λ}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$and $\mathit{I}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$, show that $\mathit{λ}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathit{k}\mathit{t}$, $\mathit{I}\mathbf{\left(}\mathit{z}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathit{k}\mathit{z}$, where k is a constant.

(b) Assume for a moment that the process is quasistatic, so the fields are given by Eqs. 2.9 and 5.38. Show that these are in fact the exact fields, by confirming that all four of Maxwell's equations are satisfied. (First do it in differential form, for the region s > 0, then in integral form for the appropriate Gaussian cylinder/Amperian loop straddling the axis.)

Question: Suppose $\mathit{j}\mathbf{\left(}\mathit{r}\mathbf{\right)}$is constant in time but $\mathit{ÒÏ}\mathbf{(}\mathbf{r}\mathbf{,}\mathbf{t}\mathbf{)}$is not-conditions that

might prevail, for instance, during the charging of a capacitor.

(a) Show that the charge density at any particular point is a linear function of time:

$\mathit{ÒÏ}\left(r,t\right)\mathbf{=}\mathit{ÒÏ}\left(r,0\right)\mathbf{+}\mathit{ÒÏ}\left(r,0\right)\mathit{t}$

where$\mathit{ÒÏ}\mathbf{(}\mathbf{r}\mathbf{,}\mathbf{0}\mathbf{)}$is the time derivative of at . [Hint: Use the continuity equation.]

This is not an electrostatic or magnetostatic configuration: nevertheless, rather surprisingly, both Coulomb's law (Eq. 2.8) and the Biot-Savart law (Eq. 5.42) hold, as you can confirm by showing that they satisfy Maxwell's equations. In particular:

(b) Show that

$\mathit{B}\left(r\right)\mathbf{=}\frac{{\mathbf{μ}}_{\mathbf{0}}}{\mathbf{4}\mathbf{π}}\mathbf{∫}\frac{\mathbf{J}\left(r\text{'}\right)\mathbf{×}\hat{\mathbf{r}}}{{\mathbf{r}}^{\mathbf{2}}}\mathit{d}\mathit{τ}\mathbf{\text{'}}$

obeys Ampere's law with Maxwell's displacement current term.

A metal bar of mass m slides frictionlessly on two parallel conducting rails a distance l apart (Fig. 7 .17). A resistor R is connected across the rails, and a uniform magnetic field B, pointing into the page, fills the entire region.

(a) If the bar moves to the right at speed V, what is the current in the resistor? In what direction does it flow?

(b) What is the magnetic force on the bar? In what direction?

(c) If the bar starts out with speed${\mathbf{V}}_{\mathbf{0}}$at time t=0, and is left to slide, what is its speed at a later time t?

(d) The initial kinetic energy of the bar was, of course,$\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{mv}}^{\mathbf{2}}$Check that the energy delivered to the resistor is exactly $\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{mv}}^{\mathbf{2}}$.

A square loop of wire (side a) lies on a table, a distance s from a very long straight wire, which carries a current I, as shown in Fig. 7.18.

(a) Find the flux of B through the loop.

(b) If someone now pulls the loop directly away from the wire, at speed, V what emf is generated? In what direction (clockwise or counter clockwise) does the current flow?

(c) What if the loop is pulled to the right at speed V ?