1. A metal rod with resistance R, is forced to move
with constant velocity v along two parallel metal rails,
separated by a distance of d, connected with a strip of
metal at one end. A uniform magnetic field B points out
of the page. Find the (a) direction and magnitude of the
induced current in the rod and (b) the direction of the
magnetic force on the rod.
ℰ
−1 𝜕Φ
−1 𝜕
−1
𝜕
−1
𝜕𝑙
−𝐵𝑑𝑣
𝐵
(𝐵 ∙ 𝐴) = 𝐵 (𝑑𝑙) = 𝐵𝑑 =
a. 𝐼 = =
=
The magnetic flux increases out
𝑅
𝑅 𝜕𝑡
𝑅 𝜕𝑡
𝑅
𝜕𝑡
𝑅
𝜕𝑡
𝑅
of the board, as the metal rod moves, so the induced current must go CLOCKWISE, to oppose this
change.
2 2
⃗⃗ = 𝐵 𝑑 𝑣 and
b. From answer a, we know that the current is flowing up through the rod. 𝐹⃗ = 𝐼⃗𝑑 × 𝐵
𝑅
this is to the right.
2. In the figure, a circular loop moves at
constant velocity through regions where
uniform magnetic fields of the same
magnitude are directed into or out of the
page. The induced EMF at locations the
labeled locations are:
a. clockwise,
b. counterclockwise,
c. Zero.
1. There is no field here, so no change in flux, so no current (c).
2. There is no change in flux here, (B is const), so no current (c).
3. There is no change in flux here, (B is const), so no current (c).
4. Flux changes from into the board to out of the board, so the current goes clockwise
to create a magnetic field directed into the page (a).
5. There is no change in flux here, (B is const), so no current (c).
6. There is no change in flux here, (B is const), so no current (c).
7. Here the flux changes from out of the page to zero, so the current goes
counterclockwise to create a magnetic field directed out of the page (b).
8. There is no field here, so no change in flux, so no current (c).
3. A single turn rectangular loop of wire with length
a=10 cm, width b = 5 cm and resistance R  20 is
placed near an infinitely long wire carrying current i=5 A,
as shown in the figure. The distance from the long wire to
the center of the loop is r= 20cm. (a) Find the magnitude
of the magnetic flux through the loop. (b) Find the
magnitude of the current in the loop as it moves away
from the long wire with a constant speed v= 5 cm/s. (c)
Write down the direction of the current (clockwise or
counterclockwise) induced in the loop.
22.5𝑐𝑚 𝜇0 𝑖
𝜇0 𝑖𝑎 22.5 𝑑𝑟
𝜇0 𝑖𝑎
22.5
⃗⃗ ∙ ⃗⃗⃗⃗⃗
a. Φ𝐵 = ∫ 𝐵
𝑑𝑎 = ∫17.5𝑐𝑚 2𝜋𝑟
𝑎𝑑𝑟 = 2𝜋
ln 𝑟 |17.5
= 5.9𝑥10−8 𝐴𝑚2
∫17.5 𝑟 = 2𝜋
ℇ
−1 𝜕Φ
𝜇 𝑖𝑎 𝜕
𝜇 𝑖𝑎 1 𝜕𝑟
𝜇 𝑖𝑎 1
0
0
0
b. 𝐼 = 𝑅 = 𝑅 𝜕𝑡𝐵 = 2𝜋𝑅
ln 𝑟 = 2𝜋𝑅
= 2𝜋𝑅
𝑣 = 1.25𝑥10−9 𝐴 = 1.25𝑛𝐴
𝜕𝑡
𝑟 𝜕𝑡
𝑟
c. The magnetic field from the wire (using the RHR) goes into the page above the wire. As the loop
moves away from the current carrying wire, it will have less flux into the page, so the induced
current will be CLOCKWISE in order to create a magnetic field also into the page.