**Core Concept**
Epinephrine, also known as adrenaline, is a potent vasoconstrictor that can be added to local anesthetics to prolong their duration of action. When epinephrine is combined with a local anesthetic, such as lignocaine, it causes a reduction in blood flow to the area, thereby slowing the absorption of the anesthetic from the injection site.

**Why the Correct Answer is Right**
The addition of epinephrine to lignocaine for a peripheral nerve block increases the duration of action of the local anesthetic. This is because epinephrine causes vasoconstriction, which reduces the blood flow to the area and slows down the absorption of lignocaine. As a result, the local anesthetic remains at the site of action for a longer period, prolonging its effect. The vasoconstrictive effect of epinephrine also helps to reduce the systemic toxicity of lignocaine by minimizing its absorption into the bloodstream.

**Why Each Wrong Option is Incorrect**
**Option A:** Epinephrine does not increase the risk of convulsions when added to lignocaine. In fact, the addition of epinephrine can help to reduce the risk of systemic toxicity, including convulsions, by slowing down the absorption of lignocaine.

**Option B:** The statement that epinephrine increases the risk of convulsions is incorrect, but option B is not a standalone wrong answer. However, it is combined with option A in the "C. Both (a) and (b)" option.

**Option C:** This option is incorrect because epinephrine does not increase the risk of convulsions when added to lignocaine.

**Option D:** This option is incorrect because epinephrine does indeed increase the duration of action of the local anesthetic when added to lignocaine.

**Clinical Pearl / High-Yield Fact**
When adding epinephrine to local anesthetics, it is essential to use a concentration of epinephrine that is appropriate for the specific anesthetic and the procedure being performed. This is to avoid excessive vasoconstriction, which can lead to tissue ischemia and potentially serious complications.

**✓ Correct Answer: B. Increase the duration of action of the local anesthetic**