1. Analyze Statement (I)

Statement (I): When currents vary with time, Newton's third law is valid only if momentum carried by the electromagnetic field is taken into account.

• For steady currents, magnetic forces between circuit elements can often be treated in a simple action-reaction style.
• However, when currents vary with time, the electromagnetic field itself can store and transport momentum.
• In such situations, the mechanical forces between charges/currents alone may not satisfy Newton's third law pairwise at an instant.
• To preserve overall momentum conservation, we must include field momentum.

Thus, Statement (I) is true.

2. Analyze Statement (II)

Statement (II): Ampere's circuital law does not depend on Biot-Savart's law.

Let us examine this carefully.

• In magnetostatics, Ampere's circuital law is $\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{\text{enc}}.$
• This relation can be derived from the Biot-Savart law for steady currents in suitable mathematical treatment.
• Historically and theoretically in magnetostatics, Ampere's law and Biot-Savart law are closely connected; Ampere's law is not independent in that context.
• Therefore the claim that it does not depend on Biot-Savart's law is taken as false in this exam setting.

Thus, Statement (II) is false.

3. Choose the correct option

• Statement (I): True
• Statement (II): False

So the correct option is:

$\boxed{\text{D}}$

4. Comparison with stored correct answer

Stored correct answer = D

My derived answer = D

Hence, they agree.