Question

✐ᴀ ᴍᴀɢɴᴇᴛɪᴄ ғɪᴇʟᴅ ɪɴ ᴀ ᴄᴇʀᴛᴀɪɴ ʀᴇɢɪᴏɴ ɪs ɢɪᴠᴇɴ ʙʏ ʙ = ʙᴏ ᴄᴏs(ωᴛ) ^ᴋ ᴀɴᴅ ᴀ ᴄᴏɪʟ ᴏғ ʀᴀᴅɪᴜs ᴀ ᴡɪᴛʜ ʀᴇsɪsᴛᴀɴᴄᴇ ʀ ɪs ᴘʟᴀᴄᴇᴅ ɪɴ ᴛʜᴇ x-ʏ ᴘʟᴀɴᴇ ᴡɪᴛʜ ɪᴛs ᴄᴇɴᴛʀᴇ ᴀᴛ ᴛʜᴇ ᴏʀɪɢɪɴ ɪɴ ᴛʜᴇ ᴍᴀɢɴᴇᴛɪᴄ ғɪᴇʟᴅ (sᴇᴇ ғɪɢ 6.6) . ғɪɴᴅ ᴛʜᴇ ᴍᴀɢɴɪᴛᴜᴅᴇ ᴀɴᴅ ᴛʜᴇ ᴅɪʀᴇᴄᴛɪᴏɴ ᴏғ ᴛʜᴇ ᴄᴜʀʀᴇɴᴛ ᴀᴛ (ᴀ, 0, 0) ᴀᴛ ᴛ = π/2ω, ᴛ = π/ω ᴀɴᴅ ᴛ = 3π/2ω.
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Answer 1

Question :

Question :

✐ᴀ ᴍᴀɢɴᴇᴛɪᴄ ғɪᴇʟᴅ ɪɴ ᴀ ᴄᴇʀᴛᴀɪɴ ʀᴇɢɪᴏɴ ɪs ɢɪᴠᴇɴ ʙʏ ʙ = ʙᴏ ᴄᴏs(ωᴛ) ^ᴋ ᴀɴᴅ ᴀ ᴄᴏɪʟ ᴏғ ʀᴀᴅɪᴜs ᴀ ᴡɪᴛʜ ʀᴇsɪsᴛᴀɴᴄᴇ ʀ ɪs ᴘʟᴀᴄᴇᴅ ɪɴ ᴛʜᴇ x-ʏ ᴘʟᴀɴᴇ ᴡɪᴛʜ ɪᴛs ᴄᴇɴᴛʀᴇ ᴀᴛ ᴛʜᴇ ᴏʀɪɢɪɴ ɪɴ ᴛʜᴇ ᴍᴀɢɴᴇᴛɪᴄ ғɪᴇʟᴅ (sᴇᴇ ғɪɢ 6.6) . ғɪɴᴅ ᴛʜᴇ ᴍᴀɢɴɪᴛᴜᴅᴇ ᴀɴᴅ ᴛʜᴇ ᴅɪʀᴇᴄᴛɪᴏɴ ᴏғ ᴛʜᴇ ᴄᴜʀʀᴇɴᴛ ᴀᴛ (ᴀ, 0, 0) ᴀᴛ ᴛ = π/2ω, ᴛ = π/ω ᴀɴᴅ ᴛ = 3π/2ω.

answer :

Here,

→ ^

B = Bo cos(wt)k

i.e., magnetic field is alond positive z-axis. Fig Magnetic flux linked with the ring

ϕ = BAcos0∘ =B(0)cos ωt (πa²)

emf induced in the ring,

e = - dϕ / dt = − d / dt [B0πa²cosωt]=B

current induced,

I =e / R= B0πa²ω sin ωt / R

At,

t = π / 2ω

,

I = B0πa²ω sin(π) Bπa²ω

------------ ------------- = ---------------

R 2 R

, along

^

j

At

t = π / ω

,

I = B0πa²ω

--------------sin π =

R

zero At

t = 3π / 2ω

,

I = B0πa²ω (3π) −B0πa²ω

--------------sin ------ = - -------------

R R R

, along

^

j

Note that the direction of current at any instant is as per Lenz's law.

Answer 2

Question:-

ᴀ ᴍᴀɢɴᴇᴛɪᴄ ғɪᴇʟᴅ ɪɴ ᴀ ᴄᴇʀᴛᴀɪɴ ʀᴇɢɪᴏɴ ɪs ɢɪᴠᴇɴ ʙʏ ʙ = ʙᴏ ᴄᴏs(ωᴛ) ^ᴋ ᴀɴᴅ ᴀ ᴄᴏɪʟ ᴏғ ʀᴀᴅɪᴜs ᴀ ᴡɪᴛʜ ʀᴇsɪsᴛᴀɴᴄᴇ ʀ ɪs ᴘʟᴀᴄᴇᴅ ɪɴ ᴛʜᴇ x-ʏ ᴘʟᴀɴᴇ ᴡɪᴛʜ ɪᴛs ᴄᴇɴᴛʀᴇ ᴀᴛ ᴛʜᴇ ᴏʀɪɢɪɴ ɪɴ ᴛʜᴇ ᴍᴀɢɴᴇᴛɪᴄ ғɪᴇʟᴅ (sᴇᴇ ғɪɢ 6.6) . ғɪɴᴅ ᴛʜᴇ ᴍᴀɢɴɪᴛᴜᴅᴇ ᴀɴᴅ ᴛʜᴇ ᴅɪʀᴇᴄᴛɪᴏɴ ᴏғ ᴛʜᴇ ᴄᴜʀʀᴇɴᴛ ᴀᴛ (ᴀ, 0, 0) ᴀᴛ ᴛ = π/2ω, ᴛ = π/ω ᴀɴᴅ ᴛ = 3π/2ω.

${\pink{\underline{\underline{\bold{Answer:-}}}}}$

27 ᴀ ᴍᴀɢɴᴇᴛɪᴄ ғɪᴇʟᴅ ʙ ɪs ᴄᴏɴғɪɴᴇᴅ ᴛᴏ ᴀ ʀᴇɢɪᴏɴ ʀ ≤ ᴀ ᴀɴᴅ ᴘᴏɪɴᴛs ᴏᴜᴛ ᴏғ ᴛʜᴇ ᴘᴀᴘᴇʀ (ᴛʜᴇ ᴢ-ᴀxɪs), ʀ = 0 ʙᴇɪɴɢ ᴛʜᴇ ᴄᴇɴᴛʀᴇ ᴏғ ᴛʜᴇ ᴄɪʀᴄᴜʟᴀʀ ʀᴇɢɪᴏɴ. ᴀ ᴄʜᴀʀɢᴇᴅ ʀɪɴɢ (ᴄʜᴀʀɢᴇ = ϙ) ᴏғ ʀᴀᴅɪᴜs ʙ, ʙ > ᴀ ᴀɴᴅ ᴍᴀss ᴍ ʟɪᴇs ɪɴ ᴛʜᴇ x-ʏ ᴘʟᴀɴᴇ ᴡɪᴛʜ ɪᴛs ᴄᴇɴᴛʀᴇ ᴀᴛ ᴛʜᴇ ᴏʀɪɢɪɴ. ᴛʜᴇ ʀɪɴɢ ɪs ғʀᴇᴇ ᴛᴏ ʀᴏᴛᴀᴛᴇ ᴀɴᴅ ɪs ᴀᴛ ʀᴇsᴛ. ᴛʜᴇ ᴍᴀɢɴᴇᴛɪᴄ ғɪᴇʟᴅ ɪs ʙʀᴏᴜɢʜᴛ ᴛᴏ ᴢᴇʀᴏ ɪɴ ᴛɪᴍᴇ Δᴛ. ғɪɴᴅ ᴛʜᴇ ᴀɴɢᴜʟᴀʀ ᴠᴇʟᴏᴄɪᴛʏ ω ᴏғ ᴛʜᴇ ʀɪɴɢ ᴀғᴛᴇʀ ᴛʜᴇ ғɪᴇʟᴅ ᴠᴀɴɪsʜᴇs

hope it helps you..