Question

32.2 induced magnetic fields
a changing electric flux
can not exist in the presence of a magnetic field.
may never induce a changing magnetic field.
may only induce a changing magnetic field.
may induce a constant magnetic field.

Explanation:

Step1: Recall Maxwell's equations

Maxwell - Ampere's law is \(\oint\vec{B}\cdot d\vec{l}=\mu_0\epsilon_0\frac{d\Phi_E}{dt}+\mu_0I_{enc}\). It shows the relationship between magnetic field (\(B\)) and changing electric flux (\(\frac{d\Phi_E}{dt}\)).

Step2: Analyze each option

• Option 1: A changing electric flux can exist in the presence of a magnetic field. For example, in electromagnetic waves, both electric and magnetic fields co - exist and are changing. So, the statement “can not exist in the presence of a magnetic field” is wrong.
• Option 2: From Maxwell - Ampere's law \(\oint\vec{B}\cdot d\vec{l}=\mu_0\epsilon_0\frac{d\Phi_E}{dt}+\mu_0I_{enc}\), a changing electric flux (\(\frac{d\Phi_E}{dt}

eq0\)) can induce a magnetic field. So, the statement “may NEVER induce a changing magnetic field” is wrong.

• Option 3: If \(\frac{d\Phi_E}{dt}\) is non - zero and changing (e.g., \(\frac{d\Phi_E}{dt}=kt\), \(k

eq0\)), then from \(\oint\vec{B}\cdot d\vec{l}=\mu_0\epsilon_0\frac{d\Phi_E}{dt}+\mu_0I_{enc}\), \(B\) will be changing. But if \(\frac{d\Phi_E}{dt}\) is a non - zero constant (e.g., \(\frac{d\Phi_E}{dt}=k\), \(k
eq0\)), then \(B\) will be constant. So, the statement “may ONLY induce a changing magnetic field” is wrong.

• Option 4: When \(\frac{d\Phi_E}{dt}\) is a non - zero constant (from \(\oint\vec{B}\cdot d\vec{l}=\mu_0\epsilon_0\frac{d\Phi_E}{dt}+\mu_0I_{enc}\)), \(B\) can be a non - zero constant. For example, if we have a parallel plate capacitor being charged with a constant current \(I = \epsilon_0\frac{d\Phi_E}{dt}\) (where \(I\) is constant), then from \(\oint\vec{B}\cdot d\vec{l}=\mu_0\epsilon_0\frac{d\Phi_E}{dt}+\mu_0I_{enc}\) (assuming \(I_{enc} = 0\) in the region between the capacitor plates), \(B\) is non - zero and constant.

Answer:

may induce a constant magnetic field.