When Dendritic cell travels to nearby lymph node with antigen presented on MHC II molecule, the helper T-cell residing there gets activated. But what happens to B- cell residing there? Does it get activated too when helper T-cell is activated and start producing antibodies for that specific antigen?
Naive T cells hang out in the lymph node waiting for antigen presenting cells (APCs) like DCs to present processed antigens on their MHC molecules. Naive B cells similarly hang out in the B cell zone and are capable of interacting with other T cells and follicular dendritic cells (FDCs). Antigen that ends up in the lymph ends up traveling to the lymph node, where B cells internalize, process and display them on MHC molecules. This is because B cells are also APCs.
Activated (and differentiated) T cells that were previously activated by DCs can identify some of these B cells clones through the TCR-MHC interaction. The B cells process the antigens like DCs, so the resulting heterogeneity of the processed peptides means a fair chance a T cell will recognize it. The result is that B cells receive an activating signal through the BCR and costimulatory receptors.
Differentiated T cells also secrete different profiles of cytokines. For example, Th1 cells secrete IL-12 and IFN-y whereas Th2 cells secrete IL-4, IL-5 and IL-13. Not exclusively, just predominantly or in greater amounts.
The receptor signaling in conjunction with certain cytokines can induce class switching to the resulting antibody production. This swaps the constant or Fc region of the antibody to a different isoform. So Naive B cells are IgM/IgD, and if the cells is receiving Th2 cytokine stimulation, class swtiching results in IgE antibodies, for example.
For elementary immunology I like to reference Cellular and Molecular Immunology, 8th ED. by Abbas et al., but 9th ED is out so keep that in mind. The field somewhat rapidly evolves.
There are a couple ways the adaptive immune system can be activated: Naive T cells require a co-stimulatory signal from professional antigen-presenting cells (Dendritic Cell); naive B cells require accessory signals that can come either from an armed helper T cell or, in some cases, directly from microbial antigens(NCBI, B-cell activation by armed helper T cells). So, within your question the helper T cell would activate naïve B cells through T cell-dependent activation. Though T cell-independent activation is possible (Wikipedia, B cell).
The activated B-cell would then begin producing antibodies while also undergoing clonal expansion to produce a foci of B cells that are specific for the antigen. Most of these clones differentiate into the plasma cells, also called effector B cells which produce a first wave of protective antibodies and help clear the infection, but a fraction persist as dormant memory cells that survive in the body on a long-term basis after having gone through a highly mutative and selective germinal center reaction (NCBI, Human memory B cells; Wikipedia, memory B cells).
Great visuals if you look at Google images "t dependent b cell activation".